advancements in latent space network modelling · second, we develop a latent space model for...

Advancements in Latent

Space Network Modelling

Kathryn Ruth Turnbull, MMath (Hons), MRes

Submitted for the degree of Doctor of

Philosophy at Lancaster University.

December 2019

Abstract

The ubiquity of relational data has motivated an extensive literature on network anal-

ysis, and over the last two decades the latent space approach has become a popular

network modelling framework. In this approach, the nodes of a network are repre-

sented in a low-dimensional latent space and the probability of interactions occurring

are modelled as a function of the associated latent coordinates. This thesis focuses

on computational and modelling aspects of the latent space approach, and we present

two main contributions.

First, we consider estimation of temporally evolving latent space networks in which

interactions among a fixed population are observed through time. The latent coordi-

nates of each node evolve other time and this presents a natural setting for the ap-

plication of sequential monte carlo (SMC) methods. This facilitates online inference

which allows estimation for dynamic networks in which the number of observations

in time is large. Since the performance of SMC methods degrades as the dimension

of the latent state space increases, we explore the high-dimensional SMC literature to

allow estimation of networks with a larger number of nodes.

Second, we develop a latent space model for network data in which the interactions

occur between sets of the population and, as a motivating example, we consider a

coauthorship network in which it is typical for more than two authors to contribute

to an article. This type of data can be represented as a hypergraph, and we extend the

latent space framework to this setting. Modelling the nodes in a latent space provides

a convenient visualisation of the data and allows properties to be imposed on the

I

II

hypergraph relationships. We develop a parsimonious model with a computationally

convenient likelihood. Furthermore, we theoretically consider the properties of the

degree distribution of our model and further explore its properties via simulation.

Acknowledgements

I would like to thank my supervisors Christopher Nemeth, Simon Lunagomez and

Matthew Nunes for their support and guidance throughout my PhD. Your encour-

agement has helped me to grow as a researcher and has been instrumental to the

completion of this thesis. Thank you also to Edoardo Airoldi who contributed to the

work in Chapter 4 and my external supervisor Tyler McCormick.

My PhD has been funded through the EPSRC STOR-i CDT and I would like to

thank all of those involved in STOR-i. I feel very privileged to have been able to

complete my PhD in such a dynamic, friendly and supportive environment which has

awarded me many exciting opportunities and experiences. I would especially like to

thank the directors and support staff for making STOR-i possible.

I am also thankful for the support and kindness my friends have given me through-

out my time at Lancaster. In particular, I would like to mention Lucy, Sean, Jake,

Harry, Grace, Argenis, my STOR-i cohort and the friends I have made from LLC.

Thank you especially to Wojtek for your seemingly endless supply of patience, support

and encouragement. It would not have been the same without you.

Finally, thank you to my family. You have supported me in all my endeavours

and I am eternally grateful for all that you have done for me.

III

Declaration

I declare that the work in this thesis has been done by myself and has not been

submitted elsewhere for the award of any other degree.

Kathryn Ruth Turnbull

IV

Contents

Abstract I

Acknowledgements III

Declaration IV

Contents VIII

List of Figures XIV

List of Tables XV

List of Abbreviations XVI

List of Symbols XVII

1 Introduction 1

1.1 Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Statistical Network Modelling . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Contributions and Thesis Outline . . . . . . . . . . . . . . . . . . . . 7

2 Review of Latent Space Network Modelling 10

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Latent Space Network Modelling . . . . . . . . . . . . . . . . . . . . 11

2.2.1 Generic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2.2 Distance and Projection Model . . . . . . . . . . . . . . . . . 12

V

CONTENTS VI

2.2.3 Inference and Computation . . . . . . . . . . . . . . . . . . . 14

2.2.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Exploration of Properties . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 Effect of Metric and fU . . . . . . . . . . . . . . . . . . . . . . 20

2.3.2 Effect of Covariance for Normally Distributed U . . . . . . . . 21

3 Sequential Monte Carlo and Dynamic Latent Space Networks 23

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Sequential Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 State Space Model . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.2 Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.4 High-Dimensional SMC . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Dynamic Latent Space Network Modelling . . . . . . . . . . . . . . . 33

3.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3.2 SSM formulation . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3.3 Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.4 Model Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.5 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 SMC and Dynamic Latent Space Networks . . . . . . . . . . . . . . . 38

3.4.1 State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.2 State and Parameter Estimation . . . . . . . . . . . . . . . . . 49

3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5.1 Alternative Scenarios . . . . . . . . . . . . . . . . . . . . . . . 51

3.5.2 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.6 Classroom Contact Dataset . . . . . . . . . . . . . . . . . . . . . . . 54

3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Latent Space Modelling of Hypergraph Data 60

CONTENTS VII

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.1 Latent Space Network Modelling . . . . . . . . . . . . . . . . 67

4.2.2 Random Geometric Graphs . . . . . . . . . . . . . . . . . . . 69

4.2.3 Random Geometric Hypergraphs . . . . . . . . . . . . . . . . 70

4.3 Latent space hypergraphs . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.3.2 Combining k-skeletons . . . . . . . . . . . . . . . . . . . . . . 74

4.3.3 Generative Model and Likelihood . . . . . . . . . . . . . . . . 76

4.3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.4 Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.5 Posterior Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.5.1 MCMC scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.6 Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.6.1 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.6.2 Properties of a nsRGH . . . . . . . . . . . . . . . . . . . . . . 85

4.6.3 Degree Distribution for the ith Node . . . . . . . . . . . . . . 87

4.7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.7.1 Model depth comparisons . . . . . . . . . . . . . . . . . . . . 89

4.7.2 Prior Predictive vs Posterior Predictive . . . . . . . . . . . . . 95

4.7.3 Misspecification . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.8 Real data examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.8.1 Star Wars: A New Hope . . . . . . . . . . . . . . . . . . . . . 102

4.8.2 Corporate Leadership . . . . . . . . . . . . . . . . . . . . . . . 105

4.8.3 Coauthorship for Statisticians . . . . . . . . . . . . . . . . . . 109

4.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5 Conclusions and Further Work 116

CONTENTS VIII

A Appendix for ‘Sequential Monte Carlo and Dynamic Latent Space

Networks’ 121

A.1 Gradient derivation for Euclidean distance and binary observations . 121

A.2 Gradient derivation for parameter estimation . . . . . . . . . . . . . . 122

A.3 Data simulation for Section 3.5.1 . . . . . . . . . . . . . . . . . . . . 123

B Appendix for ‘Latent Space Modelling of Hypergraph Data’ 125

B.1 Bookstein coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 125

B.1.1 Bookstein coordinates in R2 . . . . . . . . . . . . . . . . . . . 125

B.1.2 Bookstein coordinates in R3 . . . . . . . . . . . . . . . . . . . 126

B.2 Modifying the Hyperedge Indicators . . . . . . . . . . . . . . . . . . . 127

B.3 Conditional Posterior Distributions . . . . . . . . . . . . . . . . . . . 128

B.3.1 Conditional posterior for µ . . . . . . . . . . . . . . . . . . . . 128

B.3.2 Conditional posterior for Σ . . . . . . . . . . . . . . . . . . . . 129

B.3.3 Conditional posterior for ψ(0)k . . . . . . . . . . . . . . . . . . 130

B.3.4 Conditional posterior for ψ(1)k . . . . . . . . . . . . . . . . . . 130

B.4 MCMC initialisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

B.5 Practicalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

B.5.1 Smallest Enclosing Ball . . . . . . . . . . . . . . . . . . . . . . 133

B.5.2 Evaluating L(U , r,ψ(1),ψ(0);hN,K) . . . . . . . . . . . . . . . 135

B.6 Proofs for Section 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

B.6.1 Proof of Proposition 4.6.1 . . . . . . . . . . . . . . . . . . . . 137

B.6.2 Proof of Lemma 4.6.1 . . . . . . . . . . . . . . . . . . . . . . . 137

B.6.3 Proof of Theorem 4.6.1 . . . . . . . . . . . . . . . . . . . . . . 139

B.7 Prior and Posterior Predictive Degree Distributions . . . . . . . . . . 139

Bibliography 142

List of Figures

1.1.1 Example of a graph with V = {1, 2, 3, 4, 5} and E = {{1, 2}, {1, 3},

{1, 4}, {3, 4}, {3, 5}, {4, 5}}. . . . . . . . . . . . . . . . . . . . . . . . 2

2.2.1 The likelihood conditional on the latent coordinates (2.2.8) is invari-

ant to distance-preserving transformations of U . All latent configu-

rations in this Figure have the same likelihood value. . . . . . . . . 15

2.3.1 Motifs considered in simulations in Section 2.3. . . . . . . . . . . . . 20

2.3.2 Summary of simulated networks for the cases described in Table 2.2.1 22

3.2.1 Depiction of the dependence structure in a SSM. Each observation is

assumed independent conditional on latent variables which follow a

first order Markov process. The dependence on model parameters θ

is assumed throughout. . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3.1 SSM for dynamic latent space networks. The observed adjacency

matrices are modelled independently conditional on the latent coor-

dinates, and the latent coordinates are modelled with a first order

Markov process. θ is a vector of static parameters. . . . . . . . . . . 34

3.4.1 Performance of SIR and APF as N increases. The ESS and average

MSE in probability are shown in the left and right panels, respectively.

For each filter, the number of particles was fixed at M = 50000. . . 40

IX

LIST OF FIGURES X

3.4.2 Performance of the independent approximation as N increases. The

ESS and average MSE in probability are shown in the left and right

panels, respectively. For each filter, the number of particles was fixed

at M = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4.3 Effect of changing γ on the log-likelihood value for different sized

networks. Left to right: network of size N = 5, 10, 20 and 30, all with

d = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.4 Performance of nudging within an SIR filter as N increases for differ-

ent values of γ. The ESS and average MSE in probability are shown

in the left and right panels, respectively. The number of particles was

fixed at M = 10000. . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.4.5 Intermediary steps {xτt,s}Ss=0 between observations yt and yt+1. . . . 47

3.4.6 Performance of GIRF as N increases for varying number of interme-

diary states S. The ESS and average MSE in probability are shown

in the left and right panels, respectively. The number of particles was

fixed at M = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.5.1 Summary of online and offline estimation, shown in blue and orange,

respectively. Throughout we have left: (S1), middle: (S2), right:

(S3). Figure 3.5.1a shows the effective sample size and Figure 3.5.1b

shows the mean square error in probability. Figure 3.5.1c shows the

ROC curves for observations 1 to T − 1, and the line y = x is shown

in red. Figure 3.5.1d shows the ROC curve for the predicted prob-

abilities at time T , and the ROC curve for the true probabilities is

shown in grey. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.2 Performance of online estimation as the dimension of the data increases. 55

3.5.3 Time taken for each filter for increasing N (left) and increasing T

(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

LIST OF FIGURES XI

3.6.1 Fitted model for a selection of pairs of students. In both plots the blue

line represents the mean, the 2.5th and 97.5th percentile are shown in

purple, and the points correspond to the observations. The top plot

shows the connection probabilities for binary data and the bottom

plot shows the rate for count data. . . . . . . . . . . . . . . . . . . 57

3.6.2 Left: ROC curve for DLSN model fitted with a dot-product (teal) and

Euclidean distance (orange) formulation. The line y = x is shown in

red. Middle: average absolute error (AAE) for binary data simulated

from predictive probabilities at time T . The colours correspond to

those in the left plot. Right: average absolute error (AAE) for integer

data simulated from predictive rates at time T . . . . . . . . . . . . 58

4.1.1 Figures 4.1.1a and 4.1.1b depict two possible hypergraphs, where a

node belongs to a hyperedge if it lies within the associated shaded

region. Figure 4.1.1c is the graph obtained by replacing hyperedges

in Figure 4.1.1a, or equivalently in Figure 4.1.1b, by cliques. The

hypergraphs in 4.1.1a and 4.1.1b cannot be recovered from 4.1.1c.

Figure 4.1.1d presents the hypergraph relationships in Figure 4.1.1a

as a bipartite graph, where an edge from a population node to a

hyperedge node indicates membership of a hyperedge. . . . . . . . . 62

4.2.1 Example of a Cech complex. Left: Br(ui) for {ui = (ui1, ui2)}7i=1

in R2. Middle: the graph obtained by taking pairwise intersections.

Right: the hypergraph obtained by taking intersections of arbitrary

order. The shaded region between nodes 3, 5 and 6 indicates a hy-

peredge of order 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.3.1 Example of a Cech complex. Left: Br2(u) for each of 6 points in R2.

Middle: Br3(u) for each of 6 points in R2. Right: ∪3k=2D(k)rk . . . . . . 75

LIST OF FIGURES XII

4.6.1 Estimate of probability of a hyperedge occurring for k = 2 (red, solid),

k = 3 (pink, dot), k = 4 (purple, dash-dot), k = 5 (grey, dash) as

a function of rk. Points were simulated from a Normal distribution

with µ = (0, 0), and Σ = I2 in (a) and Σ =(2 00 1

)in (b), and the

theoretical probability for k = 2 is plotted for each case (black, dash). 86

4.6.2 Comparison of theoretical (black, dashed) and simulated (red, solid)

degree distribution for a hypergraph with N = 20 and K = 3. Figures

4.6.2a and 4.6.2b show the degree distribution for hyperedges of order

k = 2 and k = 3, respectively. The theoretical degree distribution for

k = 3 is calculated using a monte carlo estimate of pe3 . . . . . . . . 88

4.7.1 Depiction of motifs considered in Sections 4.7 and 4.8. . . . . . . . . 92

4.7.2 Summary of hypergraphs simulated from each of the models consid-

ered in Section 4.7.1. The cases considered are summarised in Table

4.7.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.7.3 Comparison of prior and posterior predictive degree distributions for

N∗ = 10 newly simulated nodes. In each figure, the left panel shows

the prior predictive degree distribution, and the right panel shows

a qq-plot of the prior and posterior predictive degree distributions.

Figures 4.7.3a and 4.7.3b show the degree distributions for hyperedges

of order 2 and 3, respectively. . . . . . . . . . . . . . . . . . . . . . 98

4.7.4 Summary of misspecification simulation study. Left to right: average

degree distribution, number of triangles Figure 4.7.1b3, number of

Figure 4.7.1a3, number of Figure 4.7.1a4, number of Figure 4.7.1a5

and density of order 3 hyperedges. Each row corresponds to the

misspecification cases summarised in Table 4.7.2. The y and x axes

show the quantiles of the posterior and prior predictives, respectively.

The red lines correspond to y = x. . . . . . . . . . . . . . . . . . . 101

LIST OF FIGURES XIII

4.8.1 Posterior mean of latent coordinates for the Star Wars dataset for

different upper limits on ϕ. Connections in orange, blue and purple

correspond to hyperedges of order k = 2, 3 and 4, respectively. . . . 103

4.8.2 Predicted degree distributions conditional on the fitted model. Given

U , r and ϕ we simulate the connections in the hypergraph to esti-

mate the degree distribution, and the upper limit for ϕ is 0.75 the

hyperedge density in Figure 4.8.2a and 1.5 the hyperedge density in

Figure 4.8.2b. The left plots show the degree distribution for “Luke”

and the right plots show the degree distribution for “Darth Vader”.

The order 2, 3, and 4 hyperedges are shown in green, orange, and

purple, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.8.3 Posterior mean of the latent coordinates after 25000 post burn-in it-

erations. Figure 4.8.3a: observed hypergraph with K = 4. Figure

4.8.3b: graph obtained by connecting nodes if they are contained

within the same observed hyperedge. Figure 4.8.3c: hypergraph ob-

tained from representing maximal cliques in the graph by a hyperedge.

Figure 4.8.3d: simplicial hypergraph obtained by representing cliques

in the graph by a hyperedge. Connections in orange, blue and purple

correspond to hyperedges of order k = 2, 3 and 4, respectively. . . . 107

4.8.4 Predictive distributions for motif counts for N∗ = 5 newly simulated

nodes. Top row: predictives for the motifs shown in Figures 4.7.1a3,

4.7.1a4 and 4.7.1a5. Bottom row: predictives for the motifs shown in

Figures 4.7.1b1, 4.7.1b2 and 4.7.1b3. . . . . . . . . . . . . . . . . . 108

4.8.5 Fitted model for coauthorship example. . . . . . . . . . . . . . . . . 110

4.9.1 N∗ = 19 predictive subgraph counts. Left to right: motifs depicted in

Figure 4.7.1a3, 4.7.1a4, 4.7.1a5, 4.7.1b1, 4.7.1b2 and 4.7.1b3. The red

dots correspond to the observed motif counts for the newly sampled

nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

LIST OF FIGURES XIV

A.3.1 Summary of parameter estimates. . . . . . . . . . . . . . . . . . . . 124

B.1.1 Bookstein transformation in R2. Left: original coordinates. Right:

transformed Bookstein coordinates. The points highlighted in red

are mapped to (−1/2, 0) and (1/2, 0). . . . . . . . . . . . . . . . . . 126

B.5.1 The blue shaded regions correspond to Br(ui), for i = 1, 2, 3, and the

purple shaded region is the smallest enclosing ball of the points. The

statements r∗ < r and Br(u1) ∩Br(u2) ∩Br(u3) 6= ∅ are equivalent. 134

B.7.1 Comparison of prior and posterior predictive degree distributions for

hyperedges occurring between the N∗ = 10 newly simulated nodes

and the nodes of the observed hypergraph. In each figure, the left

panel shows the prior predictive degree distribution, and the right

panel shows a qq-plot of the prior and posterior predictive degree dis-

tributions. Figures B.7.1a and B.7.1b show the degree distributions

for hyperedges of order 2 and 3, respectively. . . . . . . . . . . . . 140

B.7.2 Comparison of prior and posterior predictive degree distributions for

hyperedges occurring between the N∗ = 10 newly simulated nodes

only. In each figure, the left panel shows the prior predictive degree

distribution, and the right panel shows a qq-plot of the prior and pos-

terior predictive degree distributions. Figures B.7.2a and B.7.2b show

the degree distributions for hyperedges of order 2 and 3, respectively. 141

List of Tables

2.2.1 Cases considered in Section 2.3 simulation. pij is given by (2.2.4)

where ηij = α + s(ui, uj). When the latent coordinates are normally

distributed we take Σ1 = 0.5 ( 1 00 1 ) ,Σ2 = 0.5 ( 1 0.90.9 1 ) and Σ3 =

2 ( 1 00 1 ) and for all cases we fix α = 1. . . . . . . . . . . . . . . . . 19

4.7.1 Cases for each hypergraph model considered in the model depth com-

parison study. The case numbers correspond to the labels in Figures

4.7.2a, 4.7.2b and 4.7.2c. For all cases set N = 50 and, where appro-

priate, K = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.7.2 Types of misspecification. . . . . . . . . . . . . . . . . . . . . . . . . 99

XV

List of Abbreviations

ER Erdos-Renyi

ERGM Exponential Random Graphs

SBM Stochastic Blockmodel

LSM Latent Space Model

MCMC Markov Chain Monte Carlo

MH Metropolis-Hastings

SMC Sequential Monte Carlo

RDPG Random Dot-Product Graph

SSM State Space Model

PF Particle Filter

SIR Sequential Importance Resampling

APF Auxiliary Particle Filter

GIRF Guided Intermediate Resampling Filter

ESS Effective Sample Size

MSE Mean Square Error

RGG Random Geometric Graph

RGH Random Geometric Hypergraph

nsRGH Non-Simplicial Random Geometric Hypergraph

XVI

List of Symbols

G Graph

V Node set

E Edge set

N Number of nodes

d Dimension of latent space

yij Connection between nodes i and j

Y (N ×N) adjacency matrix

ui Latent coordinate of ith node in Rd

U N × d matrix of latent coordinates

T Number of time points

Yt (N ×N) adjacency matrix at time t

Ut (N × d) matrix of latent coordinates at time t

θ Static parameters

M Hyperedge set

ek Hyperedge of order k

K Maximum hyperedge order

GN,K Space of hypergraphs on N nodes with maximum hyperedge order K

VN Set of N nodes

EN,K Set of possible hyperedges up to order K on N nodes

XVII

Chapter 1

Introduction

Data describing interactions among a population arise in a diverse range of disciplines,

and examples include social relationships among a set of individuals (Zachary (1977)),

cooperating regions of the brain (Biswal et al. (2010)) and documents citing one

another (Ji and Jin (2016)). Data of this type can be analysed as a network, and

the ubiquity of relational data (see Leskovec and Krevl (2014) and Kunegis (2013))

has motivated an extensive literature on network analysis. However, relational data

present inferential challenges due to (i) the dependence inherent to the interactions,

and (ii) the potentially large number of observations. There currently exists a diverse

literature on network analysis which allows properties of an observed network to be

characterised, and the exact nature of this depends on the application of interest.

For example, we may be interested in the importance of certain members of the

population, identifying subsets of the population which exhibit similar patterns of

connectivity, or predicting future interactions.

This chapter will provide the necessary background for the remainder of this thesis.

Section 1.1 contains basic notation and definitions for network analysis, Section 1.2

overviews the literature on statistical network analysis, and Section 1.3 contains an

outline of the thesis and the highlights the contributions.

1

CHAPTER 1. INTRODUCTION 2

1 2

3 4

5

(a) V and E.

0 1 1 1 01 0 0 0 01 0 0 1 11 0 1 0 10 0 1 1 0

(b) Adjacency matrix.

Figure 1.1.1: Example of a graph with V = {1, 2, 3, 4, 5} and E = {{1, 2}, {1, 3},

{1, 4}, {3, 4}, {3, 5}, {4, 5}}.

1.1 Notation and Definitions

Network data may be recorded in a number of forms. For example, we may observe

binary interactions such as “friends” or “not friends” in the context of social networks,

or weighted interactions describing the number of messages shared between two mem-

bers of the population in the context of communications networks. Alternatively, we

may observe interactions between two distinct groups where connections only occur

across groups. Below we will provide the basic notation for the simplest case and

comment here that these concepts can be adapted accordingly. For a comprehensive

discussion, we refer to Newman (2010).

Typically, a network is represented as a graph G = (V,E) comprised of a node set

V and edge set E, where V indexes the population and E contains the edges that are

present in G. For a graph with N nodes, we have V = {1, 2, . . . , N} and E ⊆ EN ,

where EN = {{i, j}|i, j ∈ V } and |EN | = 2N . When the pairs {i, j} are unordered, so

that {i, j} ≡ {j, i}, the graph is undirected and when the pairs are ordered the graph

is directed. An example of an undirected graph is given in Figure 1.1.1a. To express

non-binary relationships, we can also associate a set of weights ω = {ωij|{i, j} ∈ E}

with the graph G. Finally, in a simple graph, the edges {i, j} ∈ E are unique and

there are no self-ties so {i, i} /∈ E. For the rest of this section we will restrict to

simple, undirected graphs with binary connections.


A graph can also be represented by an N ×N adjacency matrix Y whose {i, j}th

entry, yij, corresponds to the state of the (i, j)th edge. For an undirected graph

yij = yji, otherwise the graph is directed. When the connections are binary, we take

yij = 1 when {i, j} ∈ E and yij = 0 otherwise and, since we assume no self ties, we

have yii = 0 ∀ i = 1, 2, . . . , N . The adjacency matrix representation of the graph in

Figure 1.1.1a is given in Figure 1.1.1b.

There are a number of properties which may be of interest when analysing an

observed network. In an undirected graph, the degree of the ith node is the number

of nodes which share an edge with the ith node, namely Degi =∑

j 6=i yij. Given an

observed network, the empirical degree distribution is given by

P (Deg = k) =#{i ∈ V |Degi = k}

N, (1.1.1)

which simply denotes the proportion of nodes which have degree exactly equal to

k. Conditional on a generative model, it is often of interest to compare (1.1.1) with

the degree distribution derived under the model which describes the probability of

observing a node with degree k.

The density of a graph is the proportion of edges which are present and is given by∑i,j yij/

(N2

)and a subgraph refers to a smaller graph contained within G. Often motif

counts, in which the number of occurrences of a specific subgraph are considered,

are of interest. As an example, there are two occurrences of the triangle subgraph

{{i, j}, {j, k}, {i, k}} in Figure 1.1.1a. Note that the triangles {1, 4, 3} and {1, 3, 4}

are not considered distinct.

1.2 Statistical Network Modelling

The prevalence of network data has motivated a broad literature on network analysis

(see Kolaczyk (2009), Barabasi and Posfai (2016) and Newman (2010)). Given an

observed network, we wish to characterise and understand the structure of the inter-

actions, and the details of this depend on the application. It is common to take a


model-based approach in which a network can be understood in terms of a specific

characteristic or property such as, for example, the generative mechanism from which

an observed network arose or the degree distribution. In the statistical literature,

inference can then be made for a given model via, for instance, estimation and predic-

tion. In this section we will briefly overview the network modelling literature, and we

refer to Salter-Townshend et al. (2012) and Goldenberg et al. (2010) for more detail

and further references.

We begin with the Erdos-Renyi (ER) random graph model (Erdos and Renyi

(1959)) in which a fixed number of edges are chosen randomly from the(N2

)possible

edges, so that all graphs with exactly Ne edges are equally likely to occur. However, a

closely related model introduced in Gilbert (1959), in which each edge in an N node

graph occurs independently with probability p, is more often referred to as the ER

random graph model in the modern literature. To sample a graph from this model,

we must specify the success probability p and sample the observations as

yij ∼ Bern(p) (1.2.1)

for {i, j} ∈ {1, 2, . . . , N |i < j}. Though these models are well understood, they

are typically insufficient for modelling networks which exhibit complex structures.

Alternative random graph models were later introduced which allow finer control on

certain aspects of a network. For example, the preferential attachment model of

Barabasi and Albert (1999) models the growth of a graph from a seed graph, and the

mechanism which governs how additional nodes join the existing graph allows control

over the degree distribution.

Exponential random graph models (ERGMs) (Frank and Strauss (1986) Frank

(1991) Wasserman and Pattison (1996)) are a popular class of random graph models.

In an ERGM the probability of a network is defined as a function of network statistics,

such as the number of triangles, and the likelihood is specified as a member of the

exponential family. ERGMs are also referred to as p∗ models and they build upon the

p1 model of Holland and Leinhardt (1981) and the p2 model of van Duijn et al. (2004).


In the p1 model, the probability of connection depends on parameters associated with

the nodes and this model allows control over the degree distribution. A special case

of this is the β model in which the ith node is assigned a parameter βi ∈ R and the

connections are modelled as

pij =eβi+βj

1 + eβi+βj(1.2.2)

yij ∼ Bern(pij) (1.2.3)

for {i, j} ∈ {1, 2, . . . , N |i < j}. The p2 model extends the p1 to the setting where the

node specific parameters are unknown and estimated as a realisation from an under-

lying distribution. ERGMs have become popular in the network modelling literature,

and we refer to Section 3.6 of Goldenberg et al. (2010) for more details.

The stochastic blockmodel (SBM) of Nowicki and Snijders (2001) was introduced

to model graphs which exhibit a community structure. In this model, it is assumed

that nodes of the network can be partitioned into G distinct groups such that the

nodes within each group have similar patterns of connectivity. More specifically, we let

zi = (zi1, zi2, . . . , ziG) ∈ {0, 1}G denote the community membership of the ith node, for

i = 1, 2, . . . , N , and we assume that zi contains exactly one nonzero entry so that zig =

1 indicates that the ith node belongs to the gth community. The symmetric matrix

Q ∈ [0, 1]G×G then defines the probability of connections forming, where the diagonal

and off-diagonal entries correspond to within and between community connections,

respectively. The connection between nodes i and j is then modelled as

yij ∼ Bern(zTi Qzj

), (1.2.4)

for {i, j} ∈ {1, 2, . . . , N |i < j}, and a generative model can be obtained by placing a

distribution on the community assignments. Important variants of the SBM include

the degree corrected SBM (Karrer and Newman (2011)) where additional parameters

are introduced to control the degree of each node, and the mixed membership SBM

(Airoldi et al. (2009)) in which nodes may belong to multiple latent classes.


The latent space framework, as introduced in Hoff et al. (2002), has proven to

be a popular network modelling approach. In this framework, low-dimensional latent

coordinates are associated with each of the nodes and the probability of a connection

forming is modelled as a function of these coordinates. The contributions of this thesis

fall within this framework and so we refer to Chapter 2 for an in depth discussion of

this approach and the surrounding literature.

Graphons are another important class of network models (Borgs and Chayes

(2017), Lovasz (2012)), and a graphon is characterised by a function W : [0, 1]2 →

[0, 1] which determines the probability of a connection forming. To sample a graph

with N nodes from a graphon we take

xi ∼ U([0, 1]) for i = 1, 2, . . . , N (1.2.5)

yij ∼ Bern(W (xi, xj)) for {i, j} ∈ {1, 2, . . . , N |i < j} (1.2.6)

A graphon generalises many existing random graph models and, as an example, the

ER model can be specified by taking W (xi, xj) = p. Graphons have been considered

in the context of identifying communities in Eldridge et al. (2016) and developing

meaningful centrality measures for uncertain networks in Avella-Medina et al. (2018).

The models highlighted so far are vertex exchangeable, meaning that the probabil-

ity of observing a given graph is invariant to relabelling of the nodes. As a consequence

of the Aldous-Hoover theorem (Aldous (1981), Hoover (1979)), vertex exchangeable

models have been shown to generate dense graphs, in which the number of edges

grows quadratically with increasing N , or empty graphs with probability 1. In recent

years an alternative class of models have been introduced in which the the edges are

exchangeable (see Dempsey et al. (2019), Crane and Dempsey (2018), Campbell et al.

(2018) and Cai et al. (2016)) and these models have been shown to express sparse

graphs in which the number of edges grows sub-quadratically with increasing N . This

property is observed in many real world graphs and so models of this class present an

important contribution to the literature.


1.3 Contributions and Thesis Outline

This thesis contains two new contributions to the network analysis literature, which

are presented in Chapters 3 and 4. These contributions focus on computational and

modelling aspects of the latent space approach for network data. A summary of the

remainder of the chapters in this thesis are provided below, and the contributions are

highlighted where appropriate.

Chapter 2: Review of Latent Space Network Modelling

This chapter overviews the existing literature on latent space network models. We

begin by introducing a generic latent space model, and view the initial models of

Hoff et al. (2002) as a special case of this formulation. Then, we discuss inference for

these models and detail the surrounding modelling literature which builds upon the

approach of Hoff et al. (2002). Finally, we conclude with a simulation study which

explores the properties of standard latent space network models.

Chapter 3: Sequential Monte Carlo and Dynamic Latent Space Networks

This chapter focuses on temporally evolving networks in which interactions among a

fixed population are observed through time. The latent space framework has been

adapted to this setting by allowing the coordinates associated with the nodes to vary

over time (Sarkar and Moore (2006), Sewell and Chen (2015b), Durante and Dunson

(2014)). Typically, posterior samples are obtained via an MCMC scheme and in this

chapter we explore the application of sequential monte carlo (SMC) in this setting.

This has yet to be considered in the literature and has two important advantages:

(i) learning the latent representation sequentially avoids the increased mixing times

associated with MCMC as the number of observations in time increases, and (ii)

SMC methods facilitate online estimation which requires a smaller computational

cost. However, SMC methods typically do not perform well as the number of nodes in

the network increases and in this chapter we explore the literature on high-dimensional


SMC to find a methodology that is appropriate for this setting. We begin with an

overview of SMC and dynamic latent space network modelling. Then, we explore

different approaches for state estimation and we conclude with simulations and real

data examples.

Chapter 4: Latent Space Modelling of Hypergraph Data

This chapter extends the latent space approach of Hoff et al. (2002) to the setting in

which interactions occur between sets of nodes. As a motivating example we consider a

coauthorship network in which nodes correspond to authors and interactions indicate

which authors have contributed to a given paper. Data of this type are most appro-

priately represented by a hypergraph, which extends the representation discussed in

Section 1.1 beyond pairwise interactions. However, the literature on hypergraph mod-

elling is less developed and hypergraphs are typically represented by a graph in which

nodes are connected if they appear in the same interaction. This results in a loss of

structural information, and we develop a latent space hypergraph model to partially

address this gap in the literature. Our model provides a convenient visualisation of

the data and allows exploration of predictive distributions. Furthermore, by repre-

senting the nodes in a latent space, we are able to take advantage of the underlying

geometry to impose desirable properties on the interactions. By relying on tools from

computational topology, we avoid the expensive likelihood calculation implied by the

construction of Hoff et al. (2002) and develop a parsimonious model. We remove the

effects of non-identifiability by drawing a novel connection with Bookstein coordinates

from shape theory. Additionally, we theoretically examine the degree distribution of

our model and explore the broader properties via simulations. Finally, we analyse

three real world datasets using our framework.


Chapter 5: Conclusions and Further Work

In this chapter we conclude the thesis and describe three directions for future work.

We consider improvements to the scalability of the methodology discussed in the

thesis, the extension to changepoint and anomaly detection, and the exploration of

alternative underlying geometries.

Chapter 2

Review of Latent Space Network

Modelling

2.1 Introduction

The latent space approach for network modelling was first introduced in Hoff et al.

(2002), and has since given rise to a rich literature. In this approach, nodes of a

network are positioned in a low-dimensional latent space and the probability of con-

nections forming are modelled as a function of the associated latent coordinates. This

presents a convenient modelling framework in which the underlying geometry imposes

properties on the networks. For example, we may specify that nodes which lie close in

Euclidean distance are more likely to be connected and, in this case, the latent space

will not only provide an intuitive visualisation of the network but also encourage tran-

sitive relationships. Additionally, latent space network models allow control of joint

distributions on subgraph counts and facilitate exploration of predictive distributions

of new nodes. Latent space models were first introduced for social networks in Hoff

et al. (2002), however they have since been applied in a range of applications. For

instance, they are used to study biological networks in Hoff (2008a), coreadership

networks in Krivitsky et al. (2009) and financial networks in Ward et al. (2013).

10

CHAPTER 2. REVIEW OF LATENT SPACE NETWORK MODELLING 11

In this chapter we review the existing literature on latent space network modelling.

The details of the approach of Hoff et al. (2002) and its variants are given in Section

2.2, and an empirical investigation of the properties of latent space network models

is presented in Section 2.3.

2.2 Latent Space Network Modelling

In Section 2.2.1 we introduce a generic latent space network model, and then in Section

2.2.2 we view the initial models of Hoff et al. (2002) as a special case of the generic

model. Sections 2.2.3 and 2.2.4 then discuss inference and modelling extensions,

respectively.

2.2.1 Generic Model

Here we outline a generic latent space model for a network on N nodes with binary

connections Y = {yij}i,j∈{1,2,...,N}, where yij = 1 indicates the presence of the edge

{i, j}. We let ui ∈ Rd denote the d-dimensional latent coordinate associated with

the ith node, for i = 1, 2, . . . , N and we model the probability of edges forming as a

function of U = {ui}Ni=1 and additional model parameters θ. Furthermore, we assume

that the edges are conditionally independent given U . Let pij = P (yij = 1|U , θ),

then, for i, j ∈ {1, 2, . . . , N}, the connections are given by

Yijiid∼ Bern(pij), (2.2.1)

pij = h(ui, uj, θ). (2.2.2)

It is common to take h() as a function that is monotone decreasing in dij = d(ui, uj)

where, for example, d() may be chosen as the Euclidean distance between ui and uj

so that nodes which are close in a Euclidean sense are more likely to be connected.

Alternative examples will be provided in the remaining sections of this chapter.

For simplicity we will also assume there are no self-ties, so that yii = 0 for i =

1, 2, . . . , N , though this is not a necessary assumption. Finally, to obtain a generative


model, we must specify a distribution on the latent coordinates. We let

uiiid∼ fU(u|θu), for i = 1, 2, . . . , N, (2.2.3)

where θu define the distributional parameters on U .

It is typical in the latent space network modelling literature to take the latent

dimension d to be equal to 2 or 3. This allows for a convenient visualisation of the

data, however the choice of d will affect the flexibility of the model and several authors

have considered methods for choosing d. For example, Durante and Dunson (2014)

specify a maximum number for d that is generally larger than 3 and use a prior to

encourage only a small number of non-zero latent dimensions. This allows the data

to suggest a larger number of latent dimensions when appropriate. Alternatively,

Handcock et al. (2007) suggest that the choice of d may be thought of as a model

selection problem.

2.2.2 Distance and Projection Model

Hoff et al. (2002) introduced two models for network data, namely the distance and

projection model, and in this section we will specify these models as an instance of

the framework given in Section 2.2.1. The models of Hoff et al. (2002) also provide

the necessary context for the extensions discussed in Section 2.2.4.

In the distance and projection model, binary connections are modelled via logistic

regression in which

pij =1

1 + e−ηij. (2.2.4)

Then, to obtain the distance model we take

ηij = α− ‖ui − uj‖, (2.2.5)

fU(u|θu) = N (µ,Σ), (2.2.6)

where ‖ui − uj‖ denotes the Euclidean distance between ui and uj. α represents

the global tendency for edges to form in the network whereas the latent coordinates


capture the node specific tendencies. (2.2.5) has an intuitive interpretation, since

nodes which lie close together in the latent space are more likely to be connected.

Alternatively, to obtain the projection model of Hoff et al. (2002), we replace

(2.2.5) by

ηij = α +uTi uj|uj|

. (2.2.7)

Note that it is straightforward to incorporate covariate information into (2.2.5) and

(2.2.7) through adding a term of the form βT zij, where zij denotes a p-dimensional

vector of covariates on the edge {i, j}.

Although the distance and projection model take a similar form, (2.2.5) and (2.2.7)

impose different properties on the resulting networks. Since the Euclidean distance

is a metric, (2.2.5) imposes stronger constraints on the connections. In particular,

the Euclidean distance satisfies symmetry and the triangle inequality which suggest

reciprocity and transitive relationships are likely, respectively. Reciprocity means that

the edges {i, j} and {j, i} occur simultaneously, and transitivity refers to connections

in which “friends of friends are also friends”. In contrast to this, (2.2.7) does not

in general satisfy symmetry and the triangle inequality. Instead, (2.2.7) imposes

that nodes which lie in a similar direction from the origin are more likely to be

connected. To see this, note that uTi uj will approximately be positive when ui and

uj lie in the same quadrant and negative when ui and uj are in opposite quadrants.

We also observe that (2.2.7) will suggest reciprocated connections when ui and uj

are equidistant from the origin. Finally, we comment here that the dot-product is a

metric when the latent space is appropriately constrained.

The choice of latent distribution (2.2.6) reflects the intuition that nodes will have

varying levels of connectivity. For example, in the distance model, nodes which are

positioned close to µ are likely to share the greatest number of ties and these nodes

are also likely to be connected to other nodes of high degree. These differences will

be explored further in Section 2.3 via simulation.


2.2.3 Inference and Computation

Typically, latent space models are fitted via an MCMC scheme (see Hoff et al. (2002),

Krivitsky et al. (2009) and Salter-Townshend and McCormick (2017)) which requires

evaluation of the posterior at each iteration. Since the connections are modelled as

conditionally independent, a general expression for the distribution of Y conditional

on U and θ is given by

P (Y|U , θ) =∏

{i,j∈{1,2,...,N}|i 6=j}

pyijij (1− pij)1−yij , (2.2.8)

where in the symmetric case the product is over {i, j ∈ {1, 2, . . . , N}|i < j}. (2.2.8) is

the computational bottleneck when evaluating the posterior since the product contains

O(N2) terms and this scales poorly as N grows. Several authors have explored ap-

proximations of (2.2.8) to facilitate inference for increasing N . For example, Raftery

et al. (2012) introduce a case-control approximation in which the connections are

subsampled and Rastelli et al. (2018) develop an approximation by partitioning the

latent space. For a dot-product based model Durante and Dunson (2014) rely on a

Polya-gamma data augmentation scheme (Polson et al. (2013)) which allows poste-

rior samples to be obtained via Gibbs sampling. Alternatively, others have opted to

perform approximate inference at a reduced computational cost via variational Bayes

(for example, see Salter-Townshend and Murphy (2013) and Sewell and Chen (2017)).

Note that the conditional distribution (2.2.8) only depends on U through a func-

tion of U and, for the distance model (see (2.2.5)), (2.2.8) is invariant to distance

preserving transformations of U (see Figure 2.2.1). Typically, a Procrustes transform

(Borg and Groenen (1997)) is applied to the posterior samples as a post-processing

step (see Hoff et al. (2002)) to ensure the interpretability of the latent coordinates.

This transform finds the coordinates which minimise the squared difference to a set of

pre-specified reference coordinates. Some authors instead fix a subset of coordinates

(see McCormick and Zheng (2015)) and others interpret the posterior samples on the

probability space (see Durante and Dunson (2014)) and forgo the identifiability issues


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to distance-preserving transformations of U . All latent configurations in this Figure

have the same likelihood value.

associated with U .

2.2.4 Extensions

Alternative link functions and non-binary connections

In Section 2.2.2 binary connections were modelled via logistic regression, however

alternative forms of (2.2.2) have been considered in the literature. Hoff (2008a) model

the connections via probit regression and Rastelli et al. (2016) study properties of

latent variable network models in which the probability (2.2.2) is modelled via a

Gaussian distribution.

It is also straightforward to adapt the generic model in Section 2.2.1 to express

non-binary connections. For instance, to model integer connections where yij ∈ N0,

we may replace (2.2.1) with a Poisson likelihood and model the (i, j)th rate parameter

as a function of U and θ. For a particular example of this see Sewell and Chen (2016).

Node specific random effects

Krivitsky et al. (2009) introduce random effects into the linear predictor (2.2.5) to

capture each individual’s tendency to form connections. For directed edges, the linear


predictor takes the form

ηij = βxij − ‖ui − uj‖+ δi + γj, (2.2.9)

where δi and γi correspond to the propensity of the ith node to send and receive

connections, respectively. In the undirected case, the term γj in (2.2.9) is replaced by

δj and δi represents the latent ‘sociality’ of the ith node.

Modelling community structures

Several authors have considered modelling networks with community structure in the

latent space framework. Handcock et al. (2007) assume that the coordinates are

distributed according to a mixture of G Gaussian distributions, where G corresponds

to the number of communities. Hence, we let (2.2.3) take the form

fU(ui|θu) =G∑g=1

λgN (µg, σ2gId) (2.2.10)

where θu = (µ,σ,λ) for µ = (µ1, . . . , µG), σ = (σ1, . . . , σG) and λ = (λ1, . . . , λG).

The parameter λg represents the probability of each node belonging to the gth group

and, to ensure (2.2.10) is a valid probability distribution, we specify λ so that λg ≥ 0

for g = 1, 2, . . . , G and∑G

g=1 λg = 1. A similar approach was also taken in Krivitsky

et al. (2009), and Gormley and Murphy (2010) introduce a mixture of experts model

in which the mixture probabilities λ are modelled as a function of covariates.

In many applications, networks exhibit homophily in which nodes that share sim-

ilar characteristics are more likely to be connected. This property can be expressed

within the models in Section 2.2.2 through including covariates in the linear predic-

tor (2.2.4). Additionally, networks may also exhibit stochastic equivalence in which

nodes can be divided into groups such that nodes in the same group display similar

connectivity patterns. This is closely related to the community structures described

in the previous paragraph. Hoff (2008a) introduce the eigenmodel which combines

the approaches of Hoff et al. (2002) and Nowicki and Snijders (2001), which focus on


describing homophily and stochastic equivalence, respectively. In the eigenmodel, the

linear predictor is given by

ηij = α + βT zij + uTi Λuj, (2.2.11)

where ui denotes the d-dimensional of latent characteristics of the ith node, and Λ is

a d × d diagonal matrix. The non-zero entries of Λ describe the positive or negative

effect of the lth latent characteristic on the connection yij, for l = 1, 2, . . . , d. (2.2.11)

can also be adapted to model asymmetric ties as demonstrated in Hoff (2008b). Note

that the eigenmodel bears similarity to the mixed membership stochastic blockmodel

of Airoldi et al. (2009) in which connections are modelled as a function of latent group

memberships.

An alternative approach to expressing community structures is presented in Fos-

dick et al. (2019) where the authors introduce multiresolution network models. In

particular, they introduce the latent space stochastic blockmodel which mimics the

stochastic blockmodel of Holland et al. (1983) where the connection probabilities

are modelled as a function of latent group memberships. In this model, the within

community connection probabilities are determined by a latent space distance model

and the between community connection probabilities are modelled as an Erdos-Renyi

random graph.

Networks with multiple views

Multiview network data in which several types of relationships on the same set of

nodes have also been considered in the latent space framework. In Salter-Townshend

and McCormick (2017), each view is assigned a unique latent representation and

the influences between different connection types are explicitly modelled through an

association parameter. Similarly, Sweet et al. (2013) also assign a unique latent rep-

resentation to each view, however they develop a hierarchical approach to modelling

multiple views. Alternatively, Gollini and Murphy (2016) specify latent variables that


are common to each view and allow additional model parameters to capture the dif-

ferences between views. This approach is also taken in D’Angelo et al. (2019), and

D’Angelo et al. (2018) extend this to include additional sender and receiver effects for

each network view.

Extensions to the dot-product formulation

Variants of the projection model detailed in Section 2.2.2 have been considered in the

literature. Nickel (2007) first introduced the random dot product graph (RDPG) in

which the probability of a connection forming is a function of uTi uj, and generalisations

of the RDPG have also been studied in Young and Scheinerman (2007) and Ng and

Murphy (2019). Furthermore, Athreya et al. (2017) present a survey of inference on

RDPGs and Rubin-Delanchy et al. (2017) make the connection between the RDPG

and the mixed membership SBM of Airoldi et al. (2009). Finally, Rubin-Delanchy

et al. (2017) introduce a model called the generalised random dot product graph which

extends the RDPG to allow for dissasortative connections where “opposites attract”.

Temporally evolving networks

Another interesting extension arises when considering connections between a fixed

population of size N over time. This setting was first considered within the latent

space framework in Sarkar and Moore (2006), where the authors model the latent

trajectories via a Gaussian random walk. Alternatively, Durante and Dunson (2014)

have introduced a model in which the latent trajectories are modelled as a Gaussian

process. There exist many variants of these models (see Sewell and Chen (2015b),

Sewell and Chen (2017), Sewell and Chen (2016), Sewell and Chen (2015a), Friel

et al. (2016), Durante et al. (2016) and Durante et al. (2017)), and a discussion of

this literature will be provided in Chapter 3.


Case fU Metric

1 N2(0,Σ1) s(ui, uj) = uTi uj

2 N2(0,Σ1) s(ui, uj) = −‖ui − uj‖

3 U([−1.7, 1.7]2) s(ui, uj) = uTi uj

4 U([−1.7, 1.7]2) s(ui, uj) = −‖ui − uj‖

5 N2(0,Σ2) s(ui, uj) = −‖ui − uj‖

6 N2(0,Σ3) s(ui, uj) = −‖ui − uj‖

Table 2.2.1: Cases considered in Section 2.3 simulation. pij is given by (2.2.4) where

ηij = α + s(ui, uj). When the latent coordinates are normally distributed we take

Σ1 = 0.5 ( 1 00 1 ) ,Σ2 = 0.5 ( 1 0.9

0.9 1 ) and Σ3 = 2 ( 1 00 1 ) and for all cases we fix α = 1.

Alternative underlying geometries

In the literature discussed so far, the latent coordinates are assumed to lie in Euclidean

space. This choice impacts the properties of networks generated in this framework,

and in recent years several authors have explored the effect of modifying the underlying

geometry. For example, McCormick and Zheng (2015) consider modelling aggregated

relational data, in which each actor is asked “how many people with characteristic X

do you know?”, using a latent space model where the latent coordinates are assumed

to lie on the p dimensional sphere, Sp. Alternatively Krioukov et al. (2010) investigate

the effect of modelling the latent coordinates in hyperbolic space. They demonstrate

that this approach can express networks with a power law degree sequence by taking

advantage of varying density of the space. This idea has also been considered in the

context of link prediction by Kitsak et al. (2019). Finally, Smith et al. (2017) provide

an analysis of the effect of specifying the latent coordinates in Euclidean, Hyperbolic

and Elliptic geometry. They empirically explore how the degree distributions and

common network statistics change as a function of the latent geometry, and examine

how graph Laplacians may be used to identify an appropriate geometry.


(a) g1(b) g2 (c) g3 (d) g4 (e) g5

Figure 2.3.1: Motifs considered in simulations in Section 2.3.

2.3 Exploration of Properties

In this section we explore the properties of the distance and projection models from

Section 2.2.2 via simulation where we will assume d = 2 throughout. We will con-

sider the effect of varying the distribution of the latent coordinates and the choice of

s(ui, uj), where pij is given by (2.2.4) and ηij = α + s(ui, uj). Details of the choices

considered in this section are given in Table 2.2.1, where the latent coordinate dis-

tributions have been specified so that, when comparing uniform and Gaussian, the

coordinates lie within a box of roughly the same size. Throughout, α remains fixed

so that differences between networks can be attributed to choices of fU and s(ui, uj),

and we set the number of nodes to be N = 40.

We focus on the degree distribution and the distribution of motif counts, and

the motifs considered are depicted in Figure 2.3.1. In Section 2.3.1 we compare

the Euclidean metric and dot-product when the latent coordinates are uniform and

normally distributed. Then, in Section 2.3.2, we explore the effect of varying the

parameters of normally distributed latent coordinates.

2.3.1 Effect of Metric and fU

Here we compare cases 1, 2, 3 and 4 from Table 2.2.1 in which we vary the distribution

of the latent coordinates and the form of s(ui, uj). The joint distributions of the motif

counts are shown in Figure 2.3.2a and the average degree distributions are shown in

Figure 2.3.2c.


First, we notice that there is a distinct difference between the networks simulated

using Euclidean distance and the dot-product. Most notably, we see differing patterns

in the joint motif counts and, as an example, observe the correlations in the joint

distribution of g1 and g3. Additionally, the networks simulated with the dot-product

are overall much denser. We also observe that, for each choice of s(ui, uj), the networks

are denser when the points are normally distributed. This is intuitive since the choice

of parameters of the normal distribution implies that the coordinates are placed in a

region of higher density when compared to coordinates simulated from the uniform

distribution. Alternative choices of covariance can be made to alter the density of the

networks. Finally, we see that the degree distributions are generally flatter and more

skewed when s(ui, uj) is the Euclidean distance.

2.3.2 Effect of Covariance for Normally Distributed U

Here we consider the effect of changing the covariance of the latent coordinates when

they are normally distributed. Throughout, we only consider the Euclidean distance.

Case 2 from Table (2.2.1) is included for reference, and in addition to this we consider

highly correlated and more dispersed latent coordinates in cases 3 and 4, respectively.

The joint motif counts are shown in Figure 2.3.2b and the average degree distributions

are shown in Figure 2.3.2d.

We observe similar patterns in Figure 2.3.2b, however in some cases we see less

correlation in some of the cases. Cases 2 and 5 are most similar, and the the degree

distribution is slightly flatter when the coordinates are more dispersed. For all cases

we find that the degree distributions exhibit some negative skewness. The differences

in the networks for these cases reflects the intuition offered by the Euclidean distance,

since the connectivity patterns change depending on how densely positioned the latent

coordinates are. We note here that a broad range of degree distributions can be

expressed by taking mixture distributions on the latent coordinates, as in (2.2.10).


200025003000350040004500

1000

2000

3000

4000

g1

g2

20003000400050006000

1000

2000

3000

4000

g1

g3

60009000

1200015000

1000

2000

3000

4000

g1

g4

05000

100001500020000

1000

2000

3000

4000

g1

g5

20003000400050006000

200025

0030

0035

0040

0045

00

g2

g3

60009000

1200015000

200025

0030

0035

0040

0045

00

g2

g4

05000

100001500020000

200025

0030

0035

0040

0045

00

g2

g5

60009000

1200015000

200030

0040

0050

0060

00

g3

g4

05000

100001500020000

200030

0040

0050

0060

00

g3

g5

05000

100001500020000

6000

9000

1200

0

1500

0

g4

g5

case ● ● ● ●case 1 case 2 case 3 case 4

(a) Joint motif counts for cases 1,2,3 and 4.

1000

2000

3000

4000

50010

0015

0020

00

g1

g2

100020003000400050006000

50010

0015

0020

00

g1

g3

60009000

120001500018000

50010

0015

0020

00

g1

g4

01000200030004000

50010

0015

0020

00

g1g5

100020003000400050006000

1000

2000

3000

4000

g2

g3

60009000

120001500018000

1000

2000

3000

4000

g2

g4

01000200030004000

1000

2000

3000

4000

g2

g5

60009000

120001500018000

100020

0030

0040

0050

0060

00

g3

g4

01000200030004000

100020

0030

0040

0050

0060

00

g3

g5

01000200030004000

6000

9000

1200

0

1500

0

1800

0

g4g5

case ● ● ●case 2 case 5 case 6

(b) Joint motif counts for cases 2,5,6.

0.00

0.05

0.10

0 10 20 30 40degree

prob

abili

ty casecase 1case 2case 3case 4

(c) Degree distribution for cases 1,2,3 and 4.

0.000

0.025

0.050

0.075

0.100

0 10 20 30degree

prob

abili

ty casecase 2case 5case 6

(d) Degree distribution for cases 2,5 and 6.

Figure 2.3.2: Summary of simulated networks for the cases described in Table 2.2.1

Chapter 3

Sequential Monte Carlo and

Dynamic Latent Space Networks

3.1 Introduction

Network data describing interactions among a population arise in a multitude of

disciplines (see Leskovec and Krevl (2014) and Kunegis (2013)) including sociology,

biology and finance, and a range of network models have been introduced to gain

statistical insights to data of this type (see Goldenberg et al. (2010), Salter-Townshend

et al. (2012) and Kolaczyk (2009)). In many applications interactions among a fixed

population are observed over time, and this motivates the development of methodology

for temporal network data. This type of data may present practical challenges as the

number of nodes, the number of timestamps, or both, increase.

Over the past two decades, the latent space approach has proven to be a popular

modelling framework for network data (see Kim et al. (2018)). In this approach, nodes

of the network are associated with a low-dimensional latent coordinate that encodes

the propensity for edges to form. This was first introduced for static networks in Hoff

et al. (2002), and has since been extended to the temporal case (see Sarkar and Moore

(2006), Durante and Dunson (2014) and Sewell and Chen (2015b)). The latent space

23

CHAPTER 3. SMC AND DYNAMIC LATENT SPACE NETWORKS 24

framework has a number of appealing properties. For example, the latent coordinates

provide a parsimonious representation of complex data which can also be used for

visualisation, and the latent space allows exploration of predictive distributions via

simulation.

Bayesian inference for temporally evolving latent space network models is typically

facilitated by Markov Chain Monte Carlo (MCMC) since the posterior is intractable.

In this chapter, we instead consider Sequential Monte Carlo (SMC) as the inference

mechanism and the state space model formulation of Sarkar and Moore (2006) and

Sewell and Chen (2015b) provides a natural setting for SMC which has yet to be

explored in the literature. As the number of time points increases, we comment that

SMC will avoid mixing issues associated with MCMC. Furthermore, SMC facilitates

online and recursive inference, meaning that the full inference procedure is not re-

quired to be updated given additional observations. We note that SMC has been

considered in a different context for networks in Bloem-Reddy and Orbanz (2018).

The remainder of this chapter is organised as follows. An overview of SMC method-

ology is given in Section 3.2. Then, in Section 3.3, we review the existing literature

on DLSNs and introduce the state space formulation of Sarkar and Moore (2006) and

Sewell and Chen (2015b). The application of SMC to DLSNs is discussed in Section

3.4, where we pay particular attention to methodology which is appropriate for net-

works with a large numbers of nodes. Simulations and data examples are given in

Sections 3.5 and 3.6, respectively. Finally, we conclude with a discussion in Section

3.7.

3.2 Sequential Monte Carlo

Sequential Monte Carlo (SMC) methods are a broad class of simulation-based algo-

rithms designed to estimate a posterior distribution. Although they may be used

in alternative settings, our focus will be on particle filtering for state space models


Xt−1

Yt−1

. . . Xt

Yt

Xt+1

Yt+1

. . .

Figure 3.2.1: Depiction of the dependence structure in a SSM. Each observation is

assumed independent conditional on latent variables which follow a first order Markov

process. The dependence on model parameters θ is assumed throughout.

(SSMs). For a more general overview of SMC methodology see Doucet and Johansen

(2008), Lopes and Tsay (2011) and Doucet et al. (2001). We begin in Section 3.2.1

by reviewing SSMs and we then discuss state and parameter estimation in Sections

3.2.2 and 3.2.3, respectively. Finally, we briefly discuss the SMC literature for high-

dimensional state spaces in Section 3.2.4.

3.2.1 State Space Model

SSMs are a general class of models for time series data. They model a sequence of

observations {yt}Tt=1 as conditionally independent given a sequence of latent variables

{xt}Tt=0 that are assumed to follow a first order Markov process. We let gθ(yt|xt)

represent the likelihood of each observation conditional on its corresponding latent

variable, and fθ(xt|xt−1) represent the latent transition density. Note that both den-

sities depend on additional model parameters θ. A general SSM may be defined by

the following equations, and a depiction of the dependence structure is given in Figure

3.2.1.

X0 ∼ µθ(x0).

Xt|Xt−1 = xt−1 ∼ fθ(xt|xt−1), for t = 1, 2, . . . , T, (3.2.1)

Yt|Xt = xt ∼ gθ(yt|xt), for t = 1, 2, . . . , T,


3.2.2 Particle Filter

Given a sequence {yt}Tt=1, it is natural to consider which values of {xt}Tt=0 are likely to

have generated the observations. By targeting the filtering densities {pθ(xt|y1:t)}Tt=1,

the particle filter (PF) describes a framework for estimating the latent sequence.

In this section, we detail the particle filter and we comment here that there ex-

ists a related problem in which the objective is to target the smoothing densities

{pθ(xt|y1:T )}Tt=1. This is referred to as particle smoothing, and is discussed in Doucet

and Johansen (2008).

We begin by considering an expression for the filtering density at time t. By Bayes

theorem, we obtain

pθ(xt|y1:t) =

∫gθ(yt|xt)fθ(xt|xt−1)

p(yt|y1:t−1, θ)pθ(xt−1|y1:t−1) dxt−1

∝∫gθ(yt|xt)fθ(xt|xt−1)pθ(xt−1|y1:t−1) dxt−1.

(3.2.2)

The expression (3.2.2) makes explicit the dependence between the filtering density

at time t and the filtering density at time t−1. When the transition density fθ(xt|xt−1)

and conditional likelihood gθ(yt|xt) are Gaussian, it is possible to derive an exact

expression for (3.2.2) and this results in the well known Kalman filter (Kalman (1960))

However, in general, there is no analytic expression for (3.2.2) and we instead rely on

approximations of the filtering densities.

A particle filtering scheme relies on importance sampling (IS) at each time step

to sequentially approximate the filtering densities. IS is a monte carlo method for

approximating a density of the form p(x) = p(x)/Z, where p(x) is the unnormalised

density and Z =∫p(x) dx is the normalising constant. In IS, we introduce a proposal

density q(x) and define the unnormalised weights as w(x) = p(x)/q(x). Given this,

we may write

p(x) =p(x)

Z=w(x)q(x)

Z, (3.2.3)

and Z =

∫p(x) dx =

∫w(x)q(x) dx. (3.2.4)


Now suppose we have drawn M samples {x(i)}Mi=1 from q(x). Using (3.2.3) and (3.2.4)

we can then approximate p(x) using

p(x) =M∑i=1

W (i)δx(i)(x(i)), (3.2.5)

W (i) =w(x(i))∑Mi=1w(x(i))

. (3.2.6)

The normalised weightsW (i) follow from approximating (3.2.4) by Z =∑M

i=1w(x(i))/M

and the set {x(i), w(x(i))}Mi=1 is then referred to as a particle approximation of p(x).

Given (3.2.5) and (3.2.6), we may now outline a generic particle filter. The depen-

dence highlighted in (3.2.2) suggests a recursive scheme in which an approximation of

pθ(xt−1|y1:t−1) is updated to generate an approximation of pθ(xt|y1:t). More precisely,

suppose that we have a particle approximation to pθ(xt−1|y1:t−1) which we denote

by {x(i)t−1, w(i)t−1}Mi=1, where x

(i)t−1 and w

(i)t−1 denote the particles are associated weights,

respectively. Then, to estimate pθ(xt|y1:t), we sample new particles from a proposal

distribution qθ(·|yt, xt−1), and adjust the weights accordingly. This procedure can

then be repeated for each time step. For this to work in practice, we also need to

introduce a resampling step into this scheme. Otherwise we will observe particle de-

generacy in which the weights concentrate onto a single particle, resulting in poor

quality approximations. Intuitively, the resampling step replicates more informative

particles and forgets less informative particles.

Algorithm 1 details a generic particle filter in which particles are propagated

according to qθ(xt|yt, xt−1). To obtain the best performance in terms of the vari-

ance of the importance weights (Doucet et al. (2000)) we should take the proposal

qθ(xt|yt, xt−1) to be equal to

pθ(xt|yt, xt−1) =gθ(yt|xt)fθ(xt|xt−1)

p(yt|xt−1)=

gθ(yt|xt)fθ(xt|xt−1)∫gθ(yt|xt)fθ(xt|xt−1) dxt

. (3.2.7)

However, for many practical applications it is not possible to find an exact expression

for (3.2.7). Instead, we may obtain the standard SIR filter of Gordon et al. (1993)

from Algorithm 1 by taking qθ(·|xt−1, yt) = fθ(·|xt−1) and wt = gθ(yt|·). This scheme


Algorithm 1 General Particle Filter

• Iteration t = 0:

Sample M particles {x(i)0 }Mi=1 from µθ(·) and assign weights w(i)0 = 1/M .

• Iteration t = 1, . . . , T :

Assume particles {x(i)t−1}Mi=1 with weights {w(i)t−1}Mi=1 that approximate pθ(xt−1|y1:t−1).

a) Sample a parent index a(i)t−1 for the ith particle according to the weights Wt−1 =

(w(1)t−1, w

(2)t−1, . . . , w

(M)t−1 ). Denote the sampling mechanism by a

(i)t−1 ∼ F(·|Wt−1).

b) Propagate the particles according to x(i)t ∼ qθ(·|yt, x

a(i)t−1

t−1 ). Set x(i)1:t = (x

a(i)t−1

1:t−1, x(i)t )

c) Calculate the weights using w(i)t ∝

pθ(x(i)1:t, y1:t)

qθ(x(i)t |yt, x

a(i)t−1

1:t−1)pθ(xa(i)t−1

1:t−1, y1:t−1)and normalise.

propagates particles blindly according to the transition density and, whilst this will

not perform optimally, it is straightforward to implement for a variety of models. Al-

ternatively, we may approximate p(xt|yt, xt−1) as in the auxiliary particle filter of Pitt

and Shephard (1999). Here, particles are first resampled according to the predictive

density ζ(i)t ∝ w

(i)t−1∫gθ(yt|xt)fθ(xt|x(i)t−1) dxt = w

(i)t−1p(yt|x

(i)t−1) and then propagated

via the proposal qθ(xt|yt, xt−1). Since the proposal incorporates information about

the observation yt, the APF is expected to improve upon the blind propagation in

the SIR filter. Details of the APF are given in Algorithm 2 where ξ(i)t approximates

ζ(i)t . When fθ(xt|xt−1) is Gaussian and gθ(yt|xt) is log-concave, Pitt and Shephard

(1999) suggest approximating p(yt|x(i)t−1) using a Taylor expansion. Note that in Algo-

rithm 2 we provide the implementation of Carpenter et al. (1999) who avoid an extra

resampling step included in Pitt and Shephard (1999).

Implementation of a PF requires the user to choose a resampling scheme, denoted

by F(·|Wt−1) in Algorithm 1, and standard choices include multinomial, systematic,

residual and stratified resampling (see Section 2 of Douc and Cappe (2005) for the

algorithmic details). It has been shown that systematic resampling performs com-

parably to other schemes for a range of scenarios (Douc and Cappe (2005), Doucet

and Johansen (2008)), and so we rely on this throughout. This scheme has the added


Algorithm 2 APF filter (Pitt and Shephard (1999), Carpenter et al. (1999))

• Iteration t = 0:

Sample M particles {x(i)0 }Mi=1 from µθ(·) and assign weights w(i)1 = 1/M .

• Iteration t = 1, 2, . . . , T :

Assume particles {x(i)t−1}Mi=1 with weights {w(i)t−1}Mi=1 that approximate pθ(xt−1|y1:t−1).

a) Sample indices {k1, k2, . . . , kM} from {1, 2, . . . ,M} according to probabilities

{ξ(i)t }Mi=1.

b) Propagate the particles according to x(i)t ∼ qθ(·|x(ki)t−1, yt).

c) Weight particles according to w(i)t ∝

w(ki)t−1gθ(yt|x

(i)t )fθ(x

(i)t |x

(ki)t−1)

ξ(ki)t qθ(x

(i)t |x

(ki)t−1, yt)

and normalise.

benefit of being simple to implement, however there is currently no theoretical justi-

fication of the performance. For a discussion of resampling mechanisms see Doucet

and Johansen (2008) and Douc and Cappe (2005).

The performance of a particle filter can be assessed by considering the distribution

of the importance weights at each time point. In the optimal case, all particles have

equal weight and therefore contribute the same amount of information to the likeli-

hood. Since the weights are normalised, we may consider the variance of the weights

as a measure of the quality of the particle set where a smaller variance indicates a

better quality approximation. Alternatively, we can estimate the effective sample size

(ESS) and an approximate expression for this at time t is given by

ESS =1∑M

i=1(w(i)t )2

. (3.2.8)

Although (3.2.8) is widely used in the literature, we note that some authors suggest

other choices may be preferable (see Elvira et al. (2018)).

3.2.3 Parameter estimation

The PF schemes discussed in the previous section are designed to estimate the latent

states, and in this section we consider estimation of static parameters θ. We describe


methodology for offline and online estimation in Sections 3.2.3 and 3.2.3, respectively,

and we pay particular attention to the approaches relied upon in later sections. This

discussion is not fully comprehensive, and we refer to Kantas et al. (2009), Gao and

Zhang (2012), Kantas et al. (2015) and Lopes and Tsay (2011) for an overview of the

literature.

Offline estimation

In the Bayesian setting, Chopin et al. (2013) present a sequential but offline algorithm

for joint estimation of the state vector and parameters θ and Andrieu et al. (2010)

introduce particle MCMC methods, which avoid the calculation of complex proposal

distributions in MCMC by utilising an SMC approximation to the likelihood which

leaves the target distribution invariant. In particular, Andrieu et al. (2010) introduce

Particle Independent Metropolis Hastings (PIMH) for estimating the latent states

only, and both Particle Marginal Metropolis Hastings (PMMH) and Particle Gibbs

(PG) for estimating the latent states and the model parameters.

Alternatively, in the frequentist setting, many authors consider estimation θ via

maximum likelihood estimation and a survey of current methodology is given in Sec-

tion 5.1 of Kantas et al. (2015). In this section we will focus on the approach of

Nemeth et al. (2016) in which estimates of θ are obtained via gradient ascent. We

take

θk = θk−1 + γk∇ log p(y1:T |θ)|θ=θk−1, (3.2.9)

where γk is a sequence of decreasing steps such that∑

k γk =∞ and∑

k γ2k <∞. A

typical choice for this sequence is γk = k−α with 0.5 < α < 1.

To evaluate (3.2.9) we need an expression for the score ST = ∇ log p(y1:T |θ)|θ=θk−1.

Nemeth et al. (2016) rely on a particle approximation of the score and the details

of their approach are given in Algorithm 3 where St = ∇ log p(y1:t|θ). In the offline

setting a PF is implemented nit times and, in the ith iteration, the parameters are

given by θi.


Algorithm 3 Rao-Blackwellised Score

Initialise: set m(i)0 = 0 for i = 1, 2, . . . ,M , S0 = 0.

For t = 1, 2, . . . , T

1) Run one iteration of a PF to obtain {x(i)t }Mi=1, {a(i)t−1}Mi=1 and {w(i)

t }Mi=1

(see Algorithm 1)

2) Update the mean approximation

m(i)t = λm

(a(i)t−1)

t−1 + (1− λ)St−1 +∇ log g(yt|x(i)t ) +∇ log f(x(i)t |x

(a(i)t−1)

t−1 )

3) Update the score vector

St =∑M

i=1 x(i)t m

(i)t

End

Nemeth et al. (2016) also discuss estimation of the observed information matrix,

though we do not detail this in Algorithm 3. Furthermore, Nemeth et al. (2016)

demonstrate that their approach is more accurate than the particle learning (PL)

scheme of Carvalho et al. (2010), particularly when T is large, and that PL is more

sensitive to dependency in the parameters. Their approach has a linear computational

cost, and generalises the work of Poyiadjis et al. (2011).

Online estimation

A natural approach for estimating θ in the online setting would be to find a particle

approximation to the joint density p(xt, θ|y1:t), similar to the procedure for estimat-

ing {xt}Tt=1 outlined in Section 3.2.2. However, since θ does not evolve in time, the

particle set will degenerate under repeated resampling. Online estimation of θ there-

fore presents a challenging task and remains an open problem in the literature. Here

we discuss existing methodology, distinguishing between Bayesian and Frequentist

approaches.

In the Bayesian framework, several methods for static parameter estimation have

been proposed. For example, Gordon et al. (1993) include artificial dynamics to

reduce the degeneracy and Gilks and Berzuini (2001) rely on MCMC kernels to add


diversity to the particle set. Other approaches include practical filtering (see Polson

et al. (2008)), using kernel approximations (see Liu and West (2001)) and estimating

θ using sufficient statistics (see Storvik (2002), Fearnhead (2002) and Carvalho et al.

(2010)).

In the frequentist setting, θ can be estimated according to an expectation-maximisation

procedure (for example see Cappe (2011)) or via likelihood maximisation (for exam-

ple see Poyiadjis et al. (2011)). Here we again focus on the approach of Nemeth

et al. (2016), and this can be implemented in an online manner when the following

approximation is made.

∇ log p(yt|y1:t−1, θt) = ∇ log p(y1:t|θt)−∇ log p(y1:t−1|θt−1). (3.2.10)

Then, using the procedure in Algorithm 3, we can approximate the score ST by

∇ log p(yt| y1:t−1, θt) = St − St−1 at the tth. This allows us to perform the parameter

update given in (3.2.9).

3.2.4 High-Dimensional SMC

SMC methods have proven successful in a range of problems, however they typically

do not perform well as the dimension of the state space increases (Bengtsson et al.

(2008), Snyder et al. (2008)) and it has been shown that the number of particles must

increase exponentially with the state dimension to avoid particle degeneracy (Snyder

et al. (2015)). Developing methodology for this scenario remains an open problem in

the literature, and in recent years a body of work has developed to address this issue.

A review of recent methodology can be found in Septier and Peters (2015) where

they discuss the bridging density approach of Godsill and Clapp (2001), the block

particle filter of Rebeschini and van Handel (2015), and the space-time particle filter

of Beskos et al. (2017). Finally, Septier and Peters (2015) end their discussion with

the Sequential MCMC (SMCMC) approach (Septier et al. (2009), Khan et al. (2005),

Brockwell et al. (2012)). Alternative approaches not included in this review include


the guided intermediary particle filter of Park and Ionides (2019), gradient nudging

as detailed in Akyildiz and Mıguez (2019), and the nested particle filter of Naesseth

et al. (2015). In Section 3.4 we consider this literature in the context of dynamic

latent space network models and discuss the relevant approaches in more detail.

3.3 Dynamic Latent Space Network Modelling

In this section we consider modelling a temporally evolving network on N nodes.

We focus on the latent space approach, as discussed in Chapter 2. This framework

provides an intuitive visualisation of the network and is able to express properties

that are observed in many real world networks, such as transitivity. The properties

of latent space network models are well understood (Rastelli et al. (2016)) and there

is a broad modelling literature centred around this idea (for example see Handcock

et al. (2007), Krivitsky et al. (2009) and Kim et al. (2018)).

3.3.1 Background

We focus on the SSM approach of Sewell and Chen (2015b) and, prior to introducing

this in Section 3.3.2, we first discuss the necessary background. Our focus is on the

setting in which a network on N nodes is observed over time. Whilst we refer to

time-varying networks as dynamic networks, note that this term may also refer to

networks in which the population of nodes evolves or to processes on networks and

we do not consider these cases here.

The latent space framework was first considered for dynamic networks in Sarkar

and Moore (2006) and in this work the authors rely on a SSM to extend the model of

Hoff et al. (2002) to the dynamic setting. A similar approach was taken in Sewell and

Chen (2015b), however these works differ in their approach to model fitting. Sarkar

and Moore (2006) take an optimisation based approach, whereas Sewell and Chen

(2015b) rely on an MCMC scheme to obtain posterior samples. In this chapter, our


(Ut−1, θ)

Yt−1

. . . (Ut, θ)

Yt

(Ut+1, θ)

Yt+1

. . .

Figure 3.3.1: SSM for dynamic latent space networks. The observed adjacency matri-

ces are modelled independently conditional on the latent coordinates, and the latent

coordinates are modelled with a first order Markov process. θ is a vector of static

parameters.

focus will be on the SSM approach, which we discuss in Section 3.3.2. However, we

also note the approach explored in Durante and Dunson (2014) and related works,

where the authors model the latent trajectory of each node using a Gaussian process.

3.3.2 SSM formulation

Here we discuss the dynamic latent space network model of Sewell and Chen (2015b),

and we begin by introducing some notation. In this setting, we assume that we have

observed connections between N nodes at times t = 1, 2, . . . , T and the observed

binary connections at time t will be denoted by the (N × N) symmetric adjacency

matrix Yt. This matrix contains entries yijt = 1 if nodes i and j share a connection

at time t, and yijt = 0 otherwise. Additionally, we will assume that there are no self

ties so that yiit = 0 ∀ t = 1, 2, . . . , T and i = 1, 2, . . . , N . We denote the d-dimensional

latent coordinates of the ith node at time t by uit, and we let Ut ∈ RN×d be the N × d

matrix whose ith row corresponds to uit.

To model the temporal aspect of the data we assume that the latent coordinates

follow a first order Markov process and, conditional on the latent trajectories, each

observed adjacency matrix occurs independently. Additionally, we assume that the

latent coordinates of each node are independent of one another. This corresponds to

a SSM and a depiction of the dependence structure is given in Figure 3.3.1.

As in Hoff et al. (2002), the binary connections can be modelled using a logistic re-


gression model where the probability of connection depends on the latent coordinates.

A dynamic latent space network model can be specified as follows.

p(U0|θ) =N∏i=1

Np(0, τ 2Ip) (3.3.1)

p(Ut|Ut−1, θ) =N∏i=1

Np(ui,t−1, σ2Ip) for t = 1, 2, . . . T (3.3.2)

p(Yt|Ut, θ) =∏i<j

pyijtijt (1− pijt)1−yijt for t = 1, 2, . . . T (3.3.3)

where

pijt =1

1 + e−ηijt(3.3.4)

ηijt = α− ‖uit − ujt‖ (3.3.5)

for (i, j) ∈ {i, j ∈ {1, 2, . . . , N}|i < j}. θ = (α, σ, τ) ∈ R × R>0 × R>0 denotes the

model parameters; α controls the global tendency for a connection to form between

two nodes; σ defines the variance in the latent trajectories; τ defines the variance in

the initial latent coordinates. As in the static case, the latent representation captures

the node specific tendencies for a connection to form between each pair of nodes. This

model is the time-varying extension of the distance model of Hoff et al. (2002), and

the projection model can be extended in a similar way. The model specified in (3.3.1)

- (3.3.5) is simpler than the model introduced in Sewell and Chen (2015b), and we

choose this specification since our focus will be on the inference mechanism.

3.3.3 Identifiability

Note that (3.3.3) is a function of the distance between the latent coordinates and θ,

and so p(Yt|Ut, θ) is invariant to distance-preserving transformations of Ut. To resolve

this, many authors rely on a Procrustes transformation which finds the coordinates

U which minimise the sum of squared difference between U and some reference co-

ordinates U0. More precisely, the coordinates are given by U = arg minTU tr(U0 −

TU )T (U0 − TU). Note that the reference coordinates U0 are fixed and specified


beforehand. In the temporal setting, the Procrustes transformation can be applied

as a post-processing step as in Sewell and Chen (2015b), or as part of the inference

mechanism as in Sarkar and Moore (2006).

An alternative approach in the static case is to instead fix a set of latent coordi-

nates. Fixing a single coordinate will remove the effect of translations and fixing a

second will remove the effect of rotations. This approach was used by, for example,

McCormick and Zheng (2015). It is important to note that it is not straightforward

to directly apply this in the temporal setting. Other authors instead opt to examine

the model fit on the probability space, which does not suffer from non-identifiability

of U (see, for example, Durante and Dunson (2014)).

3.3.4 Model Fitting

In the Bayesian setting, inference for dynamic latent space networks is typically carried

out via a MH-within-Gibbs MCMC scheme (Sewell and Chen (2015b), Sewell and

Chen (2016)) or variational Bayes (Sewell and Chen (2017)). Variational methods

target a computationally cheaper approximation to the posterior, and this facilitates

feasible approximate inference for higher-dimensional temporal networks. Posterior

samples obtained via MCMC are from the true posterior, however this comes at a

greater computational cost. Alternatively in the frequentist setting, Sarkar and Moore

(2006) infer the parameters of their model using an optimisation scheme. Our focus

will be on the Bayesian approach, and we aim to avoid the approximations introduced

in variational methods.

Initialisation of the MCMC scheme requires careful consideration. Since there are

N × d× T latent coordinates to estimate for a network with T time stamps, this ini-

tialisation is important to obtain good performance out of the MCMC. Typically, the

latent coordinates are initialised using generalised multidimensional scaling (GMDS).

Classical MDS (see Cox and Cox (2000)) takes as an input the Euclidean distance

between all pairs of a collection of objects and returns a set of coordinates with the


specified distances. GMDS considers non-Euclidean measures of distance and, in the

case of networks, we use the shortest path between nodes i and j as the distance

measure. This idea was first introduced in Sarkar and Moore (2006).

To obtain posterior samples via MCMC we are required to evaluate the likelihood

at each iteration. Note that evaluation of (3.3.3) involves a calculation over all(N2

)pairs. This scales poorly as N grows, and we comment that this is more costly for a

directed network. Several authors have developed approximations of this likelihood

to improve the scalability of the MCMC. For example Raftery et al. (2012) introduce

the case-control approximation for latent space networks. This approach divides the

connections into {i, j} pairs which share a tie and those that do not. Then, relying

on an assumption of sparsity, they argue that the likelihood sum will be dominated

by terms with yij = 0. To alleviate the computational burden of this calculation they

approximate the likelihood by taking a subsample over these pairs. They note that

since the model is built around distances, they should sample the node pairs for which

yij = 0 accordingly. Instead of a random sample, they stratify the sample according to

the shortest path distances between each of the node pairs. This approximation was

also used in the dynamic case in Sewell and Chen (2015b) and modified to account

for missing data. An alternative approximation is introduced in Rastelli et al. (2018).

This approach assumes that the latent coordinates can be represented in a finite

box in Rp. This box is then divided into smaller boxes which form the basis of the

approximation. Since each latent coordinate lies in a single box, the authors note that

the centre of the boxes can be used as an approximation to the coordinate. Building

an approximation using this approach reduces the computational complexity when

the number of boxes is much less than the number of nodes. This has yet to be

explored for dynamic latent space networks.

To improve their MCMC scheme, Durante and Dunson (2014) rely on Polya-

gamma data augmentation (Polson et al. (2013)). This scheme was developed for

fully Bayesian inference in models with binomial likelihoods, and allows the latent


coordinates to be update via a Gibbs sampler. Whilst this is efficient, we note that this

is not compatible for a dynamic latent space model based on Euclidean distance. In

the model of Durante and Dunson (2014), the probability of a connection is determined

by the dot product between the latent coordinates.

3.3.5 Extensions

The model detailed in (3.3.1) - (3.3.5) can be modified in many ways. For example,

additional node-specific parameters which affect the tendency for individual nodes

to form connections can be included as in Sewell and Chen (2015b). Other au-

thors have developed models for non-binary connections and, in particular, Sewell

and Chen (2016) considered weighted connections and Sewell and Chen (2015a) con-

sidered ranked connections.

In Friel et al. (2016) the authors have extended this framework to the bipartite set-

ting in which there are two groups of nodes and connections only occur between nodes

in different groups. Their model incorporates higher-order temporal dependency and

they use this to model which company directors are associated with certain boards.

Another extension is considered in Sewell and Chen (2017), where the authors include

community structure by modelling the latent coordinates with a mixture of Gaussians.

Finally, Durante et al. (2017) model multilayer networks in which different types of

connections are observed for the same set of nodes, and Durante et al. (2016) model

populations of networks.

3.4 SMC and Dynamic Latent Space Networks

The dynamic latent space network (DLSN) model discussed in Section 3.3.2 presents

a natural setting for the application of SMC methods which has yet to be explored

in the literature. As commented in Section 3.3.4, the latent coordinates are typically

updated within an MCMC scheme (for example see Sewell and Chen (2015b)) and,


whilst this performs exact inference, the mixing times are expected to degenerate as

the dimension of the state space increases. This could be due to an increase in T or N ,

and we expect SMC methods to perform favourably when T grows. Furthermore, the

SMC framework facilitates both offline and online inference. In this section, we discuss

the practicalities of SMC methods in this context and, since SMC methods perform

poorly as the dimension of the latent state increases, we also focus on approaches

which are appropriate for increasing values of N . In Sections 3.4.1 and 3.4.2 we

consider state and parameter estimation, respectively.

3.4.1 State Estimation

Throughout this section we focus on estimation of the latent coordinates for the model

detailed in (3.3.1) - (3.3.5). We first consider standard particle filtering algorithms

and then examine a simplifying approximation which improves the performance of

standard algorithms. Finally, we consider two approaches from the high-dimensional

SMC methodology. Although there is a broad literature on particle filtering for high-

dimensional state spaces (see Section 3.2.4), we only consider methodology that is

appropriate for DLSNs.

We apply each method to networks simulated from the model detailed in (3.3.1) -

(3.3.5) and, when the number of nodes is the same, each method is applied to the same

simulated network. For all examples we take P = 2, T = 30, σ = 0.05, τ = 0.1, α =

0.2, and the number of nodes N is varied. For the simplifying approximation, we

also consider networks generated with a dot-product so that (3.3.5) is replaced by

ηijt = α + uTitujt, and for this formulation we take α = 0. To assess the performance

of each algorithm we consider the ESS and average MSE in the estimated connection

probabilities.


0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

T

ES

S/M

0 5 10 15 20 25 30

0.00

00.

010

0.02

0

T

MS

E

sir N=5apf N=5sir N=10apf N=10

sir N=20apf N=20sir N=30apf N=30

Figure 3.4.1: Performance of SIR and APF as N increases. The ESS and average

MSE in probability are shown in the left and right panels, respectively. For each

filter, the number of particles was fixed at M = 50000.

SIR and APF

It is natural to begin by considering the standard particle filters and in this section we

focus our attention on the SIR and APF filters. The SIR filter is obtained by setting

qθ(·|xt−1, yt) = fθ(·|xt−1) and wt = gθ(yt|·) in Algorithm 1 and the APF is detailed in

Algorithm 2. We expect the APF to outperform the SIR since particles at time t are

propagated according to the observation at time t+ 1. The SIR has the advantage of

being straightforward to implement, however the performance of this filter degrades

as the dimension of the state space increases.

To implement the APF filter we must resample the particles at time t according

to the predictive ζ(i)t ∝ q

(i)t−1p(yt|x

(i)t−1). For our model of interest, there is no analytic

expression for this and so we must rely on an approximation. Pitt and Shephard

(1999) suggest relying on a Taylor expansion when the transition and observation

densities are Gaussian and log-concave, respectively. Whilst our model satisfies the

first condition, the observation density (3.3.3) is not log-concave (see Section 3 of

Hoff et al. (2002)) and so this approximation is not appropriate. Instead we use an

approximation of the form pθ(yt|µt) where µt is a deterministic function of xt−1. We

take µt = E(Xt|Xt−1 = xt−1), but comment that there are alternative choices.


When estimating the latent states we note that the dependence in (3.3.3) means

that each particle must correspond to the entire set of latent coordinates. Therefore

we obtain {U (i)t , w

(i)t }Mi=1 as the particle set at time t. The dimension of the latent

state is given by N × d, and we now consider the performance of the SIR and APF as

the number of nodes N increases. It is important to note that these algorithms are

theoretically appropriate for high-dimensional state spaces, but the number of parti-

cles required to achieve good performance comes with an associated computational

cost which quickly becomes impractical. Under certain settings, Bickel et al. (2008)

show that the number of particles should scale exponentially with the dimension of

the state space.

Figure 3.4.1 compares the performance of the SIR and APF filters as N increases.

Overall we see that the APF outperforms the SIR filter. Both filters suffer from

a degradation in performance as N increases, though it is clear that the SIR filter

degrades at a faster rate. The performance can be improved by increasing M , though

this will quickly become infeasible. From this we conclude that it is only appropriate

to rely on these filters when N is small.

Independent Approximation

In Section 3.4.1 we observed that the performance of standard particle filtering al-

gorithms degrade with an increase in the dimension of the state space. Recall that,

due to the dependence in (3.3.3), each particle must correspond to the entire set of

latent coordinates U . In this section we examine an approximation which facilitates

inference via standard particle filtering methods for networks of increasing N . More

specifically, we consider the following approximations to (3.3.1), (3.3.2) and (3.3.4).

sij0 ∼ N(0, τ 2s

)(3.4.1)

sijt ∼ N(sij(t−1), σ

2s

)(3.4.2)

ηijt = α + sijt (3.4.3)


0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

T

ES

S/M

0 5 10 15 20 25 30

0.00

00.

010

0.02

0

T

MS

E

N=5, eucN=5 dpN=10, euc

N=10, dpN=20, eucN=20, dp

Figure 3.4.2: Performance of the independent approximation as N increases. The ESS

and average MSE in probability are shown in the left and right panels, respectively.

For each filter, the number of particles was fixed at M = 1000.

for t = 1, 2, . . . , N , and (i, j) ∈ {i, j ∈ {1, 2, . . . , N}|i < j}. In these equations, sijt

represents the latent ‘similarity’ between nodes i and j at time t. Note that in (3.3.5)

the linear predictor is specified as a function of the Euclidean distance between uit and

ujt, though other choices can be made here. Since the Euclidean distance is a metric,

it must satisfy positivity and the triangle inequality. Imposing these constraints onto

the latent similarities is challenging and it is likely that this approximation will be

poor. Alternatively, the dot product uTitujt is not a metric and does not satisfy these

constraints.

This approximation avoids the dependence in (3.3.3) and so we may estimate each

{sijt}Tt=1 independently. Note that we lose the latent representation when using this

approximation, however the performance of the filters will not degrade with increasing

N .

Figure 3.4.2 shows the performance of this filter as N increases for networks sim-

ulated using Euclidean distance and the dot product. The simulated data correspond

to the data used in Figure 3.4.1 when connections are determined by the Euclidean

distance. We estimate σ2s in (3.4.2) from the variance of the Euclidean distance and

dot-product of the known latent trajectories. Since each similarity is estimated inde-


pendently, increasing N does not affect the ESS. The MSE reflects the intuition that

this approximation is more appropriate for the dot product, since this imposes fewer

constraints on the connections. Although this approximation appears comparable to

the SIR and APF in Figure 3.4.1, it is not obvious how to characterise the scenarios

in which this approach is appropriate.

Nudging

Since standard particle filtering algorithms perform poorly as N increases, we must

explore alternative SMC methodology. In this section we will focus on the approach

of Akyildiz and Mıguez (2019) where nudging steps are introduced to guide particles

to regions of the state space with higher likelihoods. The authors develop this to

improve the performance of particle filtering algorithms as the dimension of the state

space increases. Here we will introduce the details of this approach and then consider

this in the context of DLSNs.

Akyildiz and Mıguez (2019) develop a procedure which can be applied to a general

particle filter, but for simplicity we introduce their methodology using the SIR filter.

For a particle filter with M particles, their algorithm specifies a number of particles

M∗ which are modified according to a nudging rule. Several nudging strategies are

suggested, including using gradient information, along with different ways of choosing

the particles to be nudged. A key contribution of their work is showing that, providing

a sufficiently small number of particles are nudged, the potentially computationally

costly step of correcting for the nudge can be avoided.

We focus on gradient based nudging in which a subset of particles are shifted

according to γ∇g(yt|xt) for some γ ∈ R>0. Let I indicate the set of particles which

will be nudged, where I ⊂ [M ] = {1, 2, . . . ,M} and |I| = M∗. Akyildiz and Mıguez

(2019) show that, when γM∗ ≤√M , the correction for the nudge step can be omitted

without affecting the convergence of the filter and we restrict to this case. The details

of the procedure are presented in Algorithm 4 within an SIR filter, and we comment


Algorithm 4 Nudged Bootstrap Particle Filter

For t ≥ 1:

Propagate:

Sample M particles {x(i)t }Mi=1 from f(·|xt−1)

Nudging :

Calculate x(i)t = x

(i)t + γ∇g(yt|xt) for i ∈ I, where I ∈ [M ] and |I| = M∗

Weight :

Weight each particle according to w(i)t ∝ g(yt|x(i)t )

Resample:

Resample and store particles

that other choices for the nudging mechanisms and for obtaining I are described in

Akyildiz and Mıguez (2019).

To implement gradient nudging for the DLSN model detailed in Section 3.3.2, we

wish to find an expression for ∇Ut log p(Yt|Ut). Since the conditional likelihood is

expressed in terms of uit, we instead find ∇ukt log p(Yt|Ut). This is given by

∇ukt log p(Yt|Ut) =∑

j∈[N ]\k

ukt − ujt‖ukt − ujt‖

{eα−‖ukt−ujt‖

1 + eα−‖ukt−ujt‖− ykjt

}, (3.4.4)

and details of this calculation can be found in Appendix A.1.

The expression (3.4.4) has an intuitive interpretation. Firstly, consider the case

when yjkt = 1 and ‖ukt−ujt‖ is large. In this case the the exponent term will be small

and so the −yjkt term dominates. This means that ukt and ujt will be moved closer

together after the gradient shift. Similarly, when yjkt = 0 but ‖ukt − ujt‖ is small

the gradient will move the coordinates further apart. Hence, the gradient will aim

to move nodes that are connected closer together and nodes that are not connected

further apart.

There are several ways in which we can apply the gradient nudge (3.4.4). Es-

sentially, we must consider all node pairs and move them closer together or further

apart as appropriate. For a given node pair we may opt to move a single coordinate


0.0 0.5 1.0 1.5 2.0

−7

−5

−3

Gamma

ll

(a) N = 5

0.0 0.5 1.0 1.5 2.0

−26

−22

−18

Gamma

ll

(b) N = 10

0.0 0.4 0.8 1.2

−16

0−

120

Gamma

ll

(c) N = 20

0.0 0.4 0.8

−28

0−

220

Gamma

ll

(d) N = 30

Figure 3.4.3: Effect of changing γ on the log-likelihood value for different sized net-

works. Left to right: network of size N = 5, 10, 20 and 30, all with d = 2.

proportional to γ. However, recall in our model that the latent trajectory of each

node follows a Gaussian random walk. Therefore we should ensure that certain nodes

are not shifted disproportionately more than others. To achieve this, we instead opt

to move each node in a node pair proportional to γ/2.

The choice of γ plays an important role in the performance of this approach. To

explore this, we examine the effect of nudging the states U by γ∇ log p(Y|U) for

different values of γ. Figure 3.4.3 shows the conditional log-likelihood of the γ shifted

coordinates, Uγ = U + γ∇ log p(Y|U), as a function of γ for networks of different

sizes. For each plot, the starting value of U was kept the same so that any change

in log-likelihood is due to γ. From this figure, we see that if γ is too large then the

coordinates are nudged beyond regions of higher likelihood, and that γ must scale

according to N . For example, comparing Figures 3.4.3a and 3.4.3c we see that the

same value of γ can correspond to an increase in log-likelihood in Figure 3.4.3a and a

decrease in log-likelihood in Figure 3.4.3c. This is intuitive since each node is nudged

according to all other N − 1 nodes. Finally, we comment that if the initial value of

U is poor, it may be difficult to improve the log-likelihood through gradient nudging.

Figure 3.4.4 shows the performance of nudging within an SIR filter for increasing

N . For each N , we apply nudging with γ = (0.1, 0.3, 0.5, 0.7, 0.9). Although the

performance is reasonable in terms of MSE, it is clear that nudging can drastically

alter the distribution of the filtering weights. Ultimately, the performance of nudg-


0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

T

ES

S/M

0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

T

MS

E

N=10, g=0.1N=10, g=0.3N=10, g=0.5N=10, g=0.7N=10, g=0.9

N=20, g=0.1N=20, g=0.3N=20, g=0.5N=20, g=0.7N=20, g=0.9

Figure 3.4.4: Performance of nudging within an SIR filter as N increases for different

values of γ. The ESS and average MSE in probability are shown in the left and right

panels, respectively. The number of particles was fixed at M = 10000.

ing will depend on the underlying filter and the plot analogous to Figure 3.4.4 for

nudging within an APF looks largely similar. To improve the stability we also con-

sidered scaling the gradient according to its magnitude, however this did not offer a

significant improvement. It is clear that the performance of nudging in this context

is unpredictable.

GIRF

We now consider the Guided Intermediate Resampling Filter (GIRF), as introduced

in Park and Ionides (2019), for high-dimensional filtering. In this work artificial

intermediary states are introduced between observations to guide particles to regions

of the state space with high probability. Here we will overview the GIRF algorithm

and consider its application to DLSNs.

Recall the general SSM (3.2.1) in which we let {yt}Tt=1 represent a sequence of

observations. The GIRF introduces S − 1 intermediary time steps between each pair

of observations {yt, yt+1}T−1t=1 and we denote these by {τt,s}Ss=0. We assume that the

latent transition density can by decomposed as

fθ(xt+1|xt) = fθ(xτt,1 |xt)fθ(xτt,2|xτt,1) . . . fθ(xt+1|xτt,S−1). (3.4.5)


xt = xτt,0

yt

. . . xτt,1 xτt,2 . . . xτt,S−1xτt,S = xt+1

yt+1

. . .

Figure 3.4.5: Intermediary steps {xτt,s}Ss=0 between observations yt and yt+1.

where {τt,s}Ss=0 satisfy τt,0 := t < τt,1 < · · · < τt,S−1 < τt,S := t + 1. Note that τt,0

and τt,S correspond to the times t and t + 1, respectively, and for a depiction of the

intermediary states see Figure 3.4.5.

At each intermediary time step the particles are weighted according to an as-

sessment function uτt,s(x) which guides particles towards future observations. This

function should be chosen so that uτ0,0(x) = 1 and uτT,0(x) = g(yT |x), and we will

discuss particular choices below. Given this function, particles at time step τt,s are

then weighted according to

ωτt,s(xτt,s , xτt,s−1) =

vτt,s(xτt,s)

vτt,s−1(xτt,s−1)if τt,s−1 6∈ 1 : T

vτt,s(xτt,s)

vτt,s−1(xτt,s−1)g(yt|xτt,s−1) if τt,s−1 ∈ 1 : T

. (3.4.6)

Using (3.4.6), we see that the likelihood `(y1:T ) = E[∏T

t=1 g(yt|xt)]

can be ap-

proximated by ˆ=∏T

t=0

∏Ss=1

1

M

∑Mi=1 ωτt,s(x

(i)τt,s , x

(i)τt,s−1).

Algorithm 5 details the GIRF for a general SSM where L = log ˆ. To implement

this procedure we must specify vτt,s(x) and the number of intermediary states S. Park

and Ionides (2019) suggest choosing vτt,s(x) ≈ p(yt+1:t+B|xtn,s = x) so that particles

are guided towards B future observations. Additionally, they show that S should

scale linearly according to the dimension of the latent states.

A connection can be drawn between the GIRF and several existing particle filtering

schemes. For example, we obtain the SIR filter as a special case if S = 1 and vτt,s =

g(yt|xτt,s). Additionally, the APF is also a special case when S = 1 and vτt,s =

g(yt|xτt,s)g(yt+1|µt+1(xτt,s)), where µt+1(xτt,s) represents a prediction of xt+1 given xτt,s .

Alternatively, we may guide the particles via a sequence of bridging densities given


Algorithm 5 Guided Intermediate Resampling Filter (GIRF)

Initialise: L = 0, x(i)τ0,0 ∼ µ(·) for i ∈ 1 : M

For t = 0 : T − 1

For s ∈ 1 : S

x(i)τt,s ∼ f(·|x(i)τt,s−1) for i ∈ 1 : M

w(i)τt,s = ωτt,s(x

(i)τt,s , x

(i)τt,s−1) for i ∈ 1 : M

L = L+ log(∑

j w(i)τt,s)

Sample b(i) such that P(b(i) = j) ∝ w(j)τt,s for i ∈ 1 : M

Set x(i)τt,s = x

(bi)τt,s

End

End

by, for example, vτt,s(x) = p(yt+1|x)s/S. This relates to annealed importance sampling

(see Neal (2001)) and has been considered in the high-dimensional filtering context

in Beskos et al. (2017), Beskos et al. (2014a) and Beskos et al. (2014b). Finally, Park

and Ionides (2019) comment that their methodology is similar to that of Del Moral

and Murray (2015), however Del Moral and Murray (2015) instead focus on the case

where observations are highly informative.

To consider the application of the GIRF to DLSNs, we must specify the form of

vτt,s(U). A straightforward option is to take vτt,s(U) = p(Yt+1|U). Alternatively, fol-

lowing Section 5 of Park and Ionides (2019), we may incorporate B future observations

by choosing

vτt,s(U) =

min{B,T−t}∏b=1

(vτt,s,τt+b(U)

)ητt,s,τt+b (3.4.7)

where vτt,s,τt+b(U) approximates pYt+b|Uτt,s (Yt+b|U) and ητt,s,τt+b controls the contribu-

tion of vτt,s,τt+b(U) in the assessment function. We take

ητt,s,τt+b = 1− (bS − s)S [(t+ b)−max(t+ b−B, 0)]

(3.4.8)

so that the contribution of observations decreases as a function of distance from

τt,s. This ensures that the potentially less accurate approximations have a smaller


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N=20, S=0.25NN=20, S=0.5NN=20, S=0.75NN=20, S=NN=30, S=0.25NN=30, S=0.5N

N=30, S=0.75NN=30, S=NN=50, S=0.25NN=50, S=0.5NN=50, S=0.75NN=50, S=N

Figure 3.4.6: Performance of GIRF as N increases for varying number of intermediary

states S. The ESS and average MSE in probability are shown in the left and right

panels, respectively. The number of particles was fixed at M = 1000.

contribution to vτt,s(U). It may be possible to approximate pYt+b|Uτt,s (Yt+b|U) via

simulation, however this is likely to be computationally expensive. Instead we take

vτt,s,τt+b(U) = g(Yt+b|mτt,s(U)), where mτt,s(U ) = E[Ut+b|Uτt,s = U

]= U , which

can be conveniently calculated.

Figure 3.4.6 shows the performance of the GIRF for different choices of S and

varying N when the assessment function is taken as vτt,s(U) = p(Yt+1|U ). There

was little difference in the plots when the assessment function was given by (3.4.7),

though it is not clear whether this is true in general. Overall, we see a much improved

performance for networks with larger N . In this example there is little difference

between the different choices of S, suggesting that just a few intermediary states vastly

improves performance of the filter even when the number of particles in relatively

small. By comparing the right plots of Figures 3.4.6 and 3.4.1, we see that the GIRF

performs much better in terms of MSE also.

3.4.2 State and Parameter Estimation

Based on the discussion in Section 3.4.1, we only consider the GIRF for the remainder

of this chapter. Here, we discuss the details of estimating static parameters via


Algorithm 6 Online parameter estimation within the GIRF

Initialise: set S0 = 0, L = 0, θ0,m(i)0 = 0 and x

(i)τ0,0 ∼ µθ0(·) for i = 1, 2, . . . ,M .

For t = 1, 2, . . . , T

1) Run S intermediary steps of the GIRF to obtain {x(i)t }Mi=1, {a(i)t−1}Mi=1 and

{w(i)t }Mi=1

(innermost loop in Algorithm 5)

2) Update the mean approximation

m(i)t = λm

(a(i)t−1)

t−1 +(1−λ)St−1+∇ log gθt−1(yt|x(i)t )+∇ log fθt−1(x

(i)t |x

(a(i)t−1)

t−1 )

3) Update the score vector

St =∑M

i=1 x(i)t m

(i)t

4) Update theta: θt = θt−1 + γk(St − St−1)

End

gradient ascent within the GIRF. We rely on the approach discussed in Section 3.2.3

and the online θ estimation scheme is outlined in Algorithm 6. For offline estimation

the θ update is applied after a full pass of the GIRF and ST replaces the term (St −

St−1) in Algorithm 6. Expressions for the gradient for the model in (3.3.1) - (3.3.5)

are give in Appendix A.2.

To set the hyperparameter τ we estimate an initial set of latent coordinates via

GMDS and set τ to be the sample variance of these coordinates. The GMDS takes as

input a distance matrix and returns a set of coordinates in Rp with the corresponding

distances. Following, Sarkar and Moore (2006) we take the path length between each

pair of nodes as the distance. σ is initialised by considering the variance between the

GMDS initialisation for the observations at t = 1 and t = 2, and α is initialised as

the value which maximises (3.3.3) for u∗ simulated according to σ and τ .


3.5 Simulations

In this section we explore the properties of the GIRF with parameter estimation, as

discussed in Section 3.4. First, we will consider the performance of this approach under

three different simulated data scenarios in Section 3.5.1. Then, we will considered the

scalability of this approach as N and T increase in Section 3.5.2.

3.5.1 Alternative Scenarios

In this section our focus will be on the accuracy of the GIRF for networks simulated

from three different scenarios. We consider simulated networks with the following

characteristics.

(S1) The nodes begin in a single group at time t = 1, evolve into two distinct groups

and return to a single group and time T .

(S2) The nodes begin in a single group at time t = 1 and then evolve into two groups.

(S3) The density of the networks change over time due to the latent representation

and α remains fixed.

To simulate from each of these scenarios, we model the latent trajectories as a

random walk that is guided by a deterministic function, similarly to how the commu-

nity memberships of the latent trajectories are modelled in Sewell and Chen (2017).

Details of this are provided in Appendix A.3, and for all cases we set N = 30, d = 2

and T = 75.

We fit the model detailed in (3.3.1) - (3.3.5) with online and offline θ estimation

to the first T − 1 observations and asses predictive accuracy for the T th observation.

Based on the simulations in Section 3.4.1 we set S = 15, and we consider the per-

formance of the guide function given in (3.4.7) and (3.4.8) with B ∈ {1, 2, 3}, where

B is the number of future observations incorporated into the guide function. When

B = 1, we take the guide function to be p(Yt+1|U).


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TP

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0.0

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TP

(d)

Figure 3.5.1: Summary of online and offline estimation, shown in blue and orange,

respectively. Throughout we have left: (S1), middle: (S2), right: (S3). Figure 3.5.1a

shows the effective sample size and Figure 3.5.1b shows the mean square error in

probability. Figure 3.5.1c shows the ROC curves for observations 1 to T − 1, and

the line y = x is shown in red. Figure 3.5.1d shows the ROC curve for the predicted

probabilities at time T , and the ROC curve for the true probabilities is shown in grey.


Figure 3.5.1 summarises the performance of the filter for each case in terms of

ESS, average MSE in probability and ROC curves. Each ROC curve compares the

proportion of true and false postives for a classifier at various thresholds, and a classi-

fier is most accurate when the ROC curve is close to the upper left hand corner of the

plot. Compared to Figure 3.4.6, Figure 3.5.1 appears to show a poorer performance

in terms of ESS and MSE. However, here we are also estimating static parameters

meaning that the inference task is more challenging and so we expect to see a slight

degradation in performance. We see that online estimation performs comparably to

offline estimation for all cases, and there is no clear advantage of incorporating mul-

tiple future observations in the guide function. The performance in terms of ROC is

comparable to the truth, and we comment that, in Figure 3.5.1d, we see a different

behaviour in case (S2) since the data belong to two communities at the T th observa-

tion. This differs to cases (S1) and (S3) where the data belong to a single group at

the T th observation.

Overall, we see that the filter performs well in a range of scenarios, and additional

improvement may be made by incorporating other structures into the model. For

example, scenarios (S1) and (S2) may be modelled with community structure and

scenario (S3) may be modelled with a temporally evolving base rate α. We also

find that estimates for σ and α are consistent between filters, though there is more

variability in the online cases. The estimates are given in Figure A.3.1 in Appendix

A.3.

3.5.2 Scalability

A key motivation for considering SMC in the context of DLSNs was the improved

scalability as the number of observations in time T increases. In this section we

explore the scalability of our choice of SMC algorithm for online estimation, and

comment that each iteration of the offline estimation procedure will scale similarly

to the online case. We consider the effects of increasing N and T separately, and


simulate data from model detailed in (3.3.1) - (3.3.5) for each case. In particular, we

simulate data according to

1. Increasing T

T ∈ {50, 100, 500, 1000}, N = 20, P = 2, α = 1.5, σ = 0.125, τ = 0.075

2. Increasing N

T = 25, N ∈ {50, 75, 100}, P = 2, α = 1.5, σ = 0.125, τ = 0.075

where, for each filter, we set S = N . For increasing T , we simulate a single dataset

with T = 1000 and apply the filter to the first t observations, where t ∈ {50, 100, 500,

1000}.

The performance of online estimation for each case is summarised in Figure 3.5.2.

Overall we see a good performance and, as expected, decreasing the number of inter-

mediary states reduces the performance. Similarly to the example in Section 3.5.1,

the performance is slighter poorer than in Section 3.4.1 due to added difficulty from

estimation of θ. The time to run each filter is given in Figure 3.5.3 and we see that

the scaling in terms of N is much worse than the scaling in terms of T . This is due

to the additional terms in the likelihood and increased number of intermediary states

required to obtain a good performance from the filter. We may reduce the computa-

tion cost by reducing S, however from Figure 3.5.2 we see that this will degrade the

performance of the filter.

3.6 Classroom Contact Dataset

In this section we consider a dataset describing face-to-face contact among primary

school children. The data record a connection if two students face each other within

a 20 second interval, and the data is available from www.sociopatterns.org and was

published in Stehle et al. (2011) and Gemmetto et al. (2014). We analyse interactions

among a class of 25 school children on an aggregate level, where we record whether


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(a) Summary of fit for increasing N and T = 25. Top: ESS for varying S. Bottom: MSE

for varying S.

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(b) Summary of fit for increasing T with N = 20 and S = 20.

Figure 3.5.2: Performance of online estimation as the dimension of the data increases.

each pair of students interacted within a 5 minute interval. If there is an interaction

between students i and j within the tth interval we set yijt = 1, otherwise we take

yijt = 0.

Similarly to Durante and Dunson (2014) we opt to assess the model fit on the

connection probabilities, though we may also obtain a visualisation of the dataset after

a procrustes transform has been applied to remove the effects of distance-preserving

transformations on the latent coordinates. We will compare the fit for the model given

in (3.3.1) - (3.3.5) with the dot-product formulation in which (3.3.5) is replaced by


50 60 70 80 90 100

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200 400 600 800

2000

6000

1000

0

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T

Tim

e (s

)

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Figure 3.5.3: Time taken for each filter for increasing N (left) and increasing T (right).

ηijt = α + uTitujt. For each model, we fit the data to the first T − 1 observations and

examine the predictives for the T th observation. Figure (3.6.1a) shows a subset of the

estimated connection probabilities for the dot-product formulation, and we see that

the connection probabilities reflect the observations well.

To further assess the fit we examine the ROC curve for the dot-product and

Euclidean distance formulation. This is shown in Figure 3.6.2a and we see that the

models perform similarly. We also assess the predictive probabilities for the T th

observation by simulating yijT according to pijT and calculating the average absolute

error (AAE) between the estimates and the observations. This is given in Figure

3.6.2b, and we see that the T th observations are predicted reasonably well.

An advantage of our approach is that it can be easily adapted to variants of the

model. We will now consider the dataset obtained by counting the number of times

students i and j interact within the tth time period, so that yijt ∈ N0. To model this

data we let pijt = e−λijtλyijtijt /yijt! and λijt = exp{α+uTitujt}, similarly to the approach

in Sewell and Chen (2016). As above, we may also consider a Euclidean formulation

in which λijt = exp{α − ‖uit − ujt‖}. The estimated rates for a subset of the data

are shown in Figure 3.6.1b and the average absolute error is given in Figure 3.6.2c.

Overall, we see that the model fits the data well and performs reasonably in terms of

predictive inference. By analysing the data in this way, we are able to obtain a finer

scaled understanding of the interactions.


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Figure 3.6.1: Fitted model for a selection of pairs of students. In both plots the

blue line represents the mean, the 2.5th and 97.5th percentile are shown in purple,

and the points correspond to the observations. The top plot shows the connection

probabilities for binary data and the bottom plot shows the rate for count data.

3.7 Discussion

In this chapter we have considered estimation of DLSNs via SMC and, since standard

SMC algorithms do not perform well as the dimension of the state space increases, we

have explored high-dimensional SMC in this context. Although there is a large litera-

ture on this topic, many proposed algorithms are not appropriate for our application

of interest. This is typically due to the dependence in the likelihood, since many

high-dimensional SMC algorithms aim to take advantage of regions of independence,

or approximate independence, within the state space (for example see Rebeschini and

van Handel (2015)). SMC methods have so far not been explored for temporally

evolving networks in the literature, however Bloem-Reddy and Orbanz (2018) rely on


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rates.

Figure 3.6.2: Left: ROC curve for DLSN model fitted with a dot-product (teal) and

Euclidean distance (orange) formulation. The line y = x is shown in red. Middle:

average absolute error (AAE) for binary data simulated from predictive probabilities

at time T . The colours correspond to those in the left plot. Right: average absolute

error (AAE) for integer data simulated from predictive rates at time T .

SMC for estimation of static networks which evolve through the addition of nodes.

This differs from our application since we focus on relationships between a fixed group

of individuals that are observed over time.

The algorithms considered in Section 3.4 can be adapted to variants of the DSLN

model given in Section 3.3.2, and we note that this is not the case for all applica-

ble SMC algorithms. Although a clear connection between the nudging approach of

Akyildiz and Mıguez (2019) and the GIRF of Park and Ionides (2019) can be drawn,

since they both aim to improve performance by shifting particles to regions of the

state space with high likelihood, we find that the GIRF has a much more stable per-

formance in our application. Furthermore, the GIRF can be viewed as a generalisation

of the SIR filter and so it is reasonable to use this approach with S = 1 when the net-

work of interest is sufficiently small as suggested by Section 3.4.1. It is also interesting

to note that, although the context differs, Bloem-Reddy and Orbanz (2018) rely on

similar SMC methodology. A range of modifications to the algorithms investigated


in Section 3.5 can also be explored and, as an example, we may incorporate nudging

steps into the GIRF. Whatever the modification, the computational cost must also

be taken into account.

In Section 3.4.1, we found that the independent approximation performed well in

terms of ESS and MSE in probability. However, it is important to stress that this only

considered state estimation and, when θ is also estimated, we find that the variance

in the ‘similarities’ is overestimated. This is a consequence of the approximation,

since the likelihood for each latent similarity no longer accounts for the effect of

neighbours of each node involved. Adapting this approximation presents an interesting

direction for further work, and this approach may also be combined with other SMC

methodology such as that of Fasano et al. (2019).

Since the increased computational cost when N increases is partly due to the ad-

ditional terms in the likelihood, it is common to rely on likelihood approximations

when the number of nodes becomes too large (see Raftery et al. (2012) and Rastelli

et al. (2018)). It would be interesting to explore this within the context of the SMC

algorithms considered in this chapter, since a direct application of these approxima-

tions may not be appropriate. An alternative direction to improve scalability may be

to consider a model similar to Fosdick et al. (2019) in which nodes are partitioned

into communities and the connection probabilities within each community are mod-

elled via a latent space. This will reduce the computational cost associated with the

likelihood.

Finally, this work may also be extended to changepoint and anomaly detection

in which the inference task is to determine the point at which the generative mech-

anism of the data changes and spurious observations, respectively. It is interesting

to consider what constitutes a change in the context of network data, and how the

framework presented here can be utilised. The latent space approach was recently

considered in the problem of anomaly detection in Lee et al. (2019), however the

authors rely on approximation inference via variational methods.

Chapter 4

Latent Space Modelling of

Hypergraph Data

4.1 Introduction

In this chapter we present a model for relational data which describe interactions

involving several members of a target population. Our focus is on modelling hyper-

graphs comprised of N nodes and M hyperedges, where a hyperedge corresponds to a

set of nodes, and we assume throughout that hyperedges are modelled randomly given

a fixed collection of labelled nodes. A common approach to modelling relationships

involving more than two nodes is to project the hypergraph onto a graph in which

the connections are assumed to occur between pairs of nodes only. Representing the

data as a graph, so that each hyperedge is replaced by a clique, allows the data to be

analysed according to an extensive graph modelling literature (see Kolaczyk (2009),

Barabasi and Posfai (2016) and Salter-Townshend et al. (2012)) which includes the

stochastic blockmodel (Holland et al. (1983)), exponential random graphs (Holland

and Leinhardt (1981)), random graph models (Erdos and Renyi (1959), Barabasi and

Albert (1999)), and latent space network models (Hoff et al. (2002)). However, rep-

resenting a hypergraph as a graph results in a loss of information (see Figure 4.1.1)

60

CHAPTER 4. LATENT SPACE MODELLING OF HYPERGRAPH DATA 61

and, although several models for hypergraphs have been introduced (see Stasi et al.

(2014), Ng and Murphy (2018), and Liu et al. (2013)), this literature is currently less

mature. Here we introduce a model for hypergraph data by considering the extension

of the latent space approach for graphs as introduced in Hoff et al. (2002). In this

framework the connections are modelled as a function of latent coordinates associated

with the nodes, and this construction has many desirable properties which we wish

to exploit when developing our model. In particular, the latent representation pro-

vides an intuitive visualisation of the graph, allows control in the joint distribution of

subgraph counts, and can encourage transitive relationships.

Hypergraph data arise in a range of disciplines (see Kunegis (2013) and Leskovec

and Krevl (2014)) including systems biology, neuroscience and marketing, and, de-

pending on the context, the interactions may have different interpretations. For ex-

ample, an interaction may indicate online communications, professional cooperation

between individuals, or dependence between random variables. As a motivating ex-

ample, consider coauthorship between academics where a connection indicates which

authors contributed to an article. We let the nodes represent the population of aca-

demics and, since multiple academics may contribute to an article, it is natural to

represent a publication by a hyperedge. Figure 4.1.1a shows a possible hypergraph

relationship between three authors, where a shaded region indicates which authors

contributed to an article. Whilst a range of inferential questions can be posed about

a hypergraph, we focus on the following.

(Q1) “Conditional on the observed relationships, how do we expect a new set of new

nodes to interact with the hypergraph?”. In the context of coauthorship, this

is equivalent to asking who additional authors will collaborate with given the

papers that have already been written. Once we have fitted a model to our data,

this question translates into a prediction problem. The model we introduce

represents the nodes of the hypergraph in a latent space, and this allows us to

explore predictive distributions by simulating new nodes in the latent space and


u1

u2u3

(a)

u1

u2u3

(b)

u1

u2

u3

(c)

u1u2u3

e1

e2

UE

(d)

Figure 4.1.1: Figures 4.1.1a and 4.1.1b depict two possible hypergraphs, where a node

belongs to a hyperedge if it lies within the associated shaded region. Figure 4.1.1c

is the graph obtained by replacing hyperedges in Figure 4.1.1a, or equivalently in

Figure 4.1.1b, by cliques. The hypergraphs in 4.1.1a and 4.1.1b cannot be recovered

from 4.1.1c. Figure 4.1.1d presents the hypergraph relationships in Figure 4.1.1a as a

bipartite graph, where an edge from a population node to a hyperedge node indicates

membership of a hyperedge.

examining their connections.

(Q2) “Which authors have a greater importance in the hypergraph?”. By associating

each node with a latent coordinate, we are able to determine a visualisation of

the hypergraph from our model. In latent space models, it is typical for the nodes

with greater degree to be placed more centrally in the latent representation.

Hence, we can determine the importance of a node by examining its latent

coordinate with respect to the full latent representation. This point will be

discussed further in Section 4.8.

To address the inferential questions (Q1) and (Q2) we develop a model for hy-

pergraph data that builds upon the latent space framework for graphs introduced in

Hoff et al. (2002). There exists a rich latent space network modelling literature (see

Krivitsky et al. (2009), Handcock et al. (2007), Friel et al. (2016), and Kim et al.

(2018)) and, although properties of these models are well understood (see Rastelli

et al. (2016)), their extension to the hypergraph setting is largely unexplored. In the


distance model of Hoff et al. (2002), nodes are more likely to be connected if their

latent coordinates lie close together in a Euclidean sense. Since the Euclidean distance

satisfies the triangle inequality, this suggests transitive relationships in which “friends

of friends are likely to be friends” are likely to occur. We wish to take advantage of

properties analogous to this in the hypergraph setting, and this leads us to consider

(Q3). In general, it is unclear how to impose properties on a hypergraph when a bi-

partite representation is used (see Figure 4.1.1d). Hence, when developing our model,

we rely on the representation shown in Figure 4.1.1a.

(Q3) “How do we formulate a latent space model for hypergraphs?”. Extending the

distance model of Hoff et al. (2002) to the hypergraph setting is non-trivial and

we wish to develop a model which uses constraints implied by the latent space

to impose properties on the hypergraphs. For instance, recall the motivating

example of coauthorship and consider the hypergraph depicted in Figure 4.1.1b.

In this example, it is intuitive that the hyperedge {u1, u2, u3} is likely to occur

given the presence of {u1, u2} and {u2, u3}. We wish to develop a model which

expresses this type of ‘higher-order transitivity’, where a set of authors to be

more likely to collaborate if a subset of them have already written a paper

together. Note that this notion of transitivity differs from other definitions of

hypergraph transitivity in the literature (for example, see Mansilla and Serra

(2008)).

We now review the existing literature on hypergraph analysis, and we begin with

work related to community detection. Several authors have considered this problem

for hypergraphs (see Chien et al. (2018), Lin et al. (2017), Kim et al. (2018), and

Ghoshdastidar and Dukkipati (2014)) where the interest is in determining the com-

munity membership of each node. To facilitate this, Ghoshdastidar and Dukkipati

(2014) have extended the stochastic blockmodel (SBM) of Holland et al. (1983) for

graphs to the k-uniform SBM in which all hyperedges are assumed to be of the same

size k. Alternatively, Zhou et al. (2006) have considered spectral clustering in the


hypergraph setting and Ke et al. (2019) have developed a tensor decomposition based

approach to determine community membership. Related to these works is the ap-

proach of Ng and Murphy (2018) who develop a model to capture clustering in the

hyperedges by extending methodology from latent class analysis (see Lazarsfeld and

Henry (1968), Goodman (1974)). Note that this differs to community detection since

the focus is on clustering structures in the hyperedges, not the nodes. Furthermore, in

the bipartite setting, Aksoy et al. (2016) consider a model for bipartite graphs which

exhibit community structure.

Link prediction for hypergraphs has also been explored in the literature by Ben-

son et al. (2018) and Sharma et al. (2014). These works consider predicting future

hyperedge connections given a sequence of time-indexed hyperedges. Benson et al.

(2018) focus on prediction of simplicial closure events in which a set of nodes ap-

pear within the same hyperedge. Their work does not rely on a formal statistical

model, but instead aims to identify which features of the hypergraph are indicative

of a simplicial closure event. Alternatively, Sharma et al. (2014) consider predicting

reoccurrence of previously observed hyperedges and they refer to this as ‘old edge’

prediction. They develop a tensor based approach to address this problem. We note

here that the prediction task we consider in (Q1) differs from these works since our

focus is on predicting connections for new nodes, and not future connections between

the observed set of nodes.

Other authors have developed models for hypergraphs by considering the exten-

sion of existing graph models. For example, Stasi et al. (2014) and Liu et al. (2013)

have developed models which allow control over the degree distribution by extending

the β-model of Holland and Leinhardt (1981) and the preferential attachment model

of Barabasi and Albert (1999), respectively. In the model of Stasi et al. (2014) each

node is assigned a parameter which controls its tendency to form connections so that

a hypergraph with certain degree distribution can be expressed. Alternatively, the

model of Liu et al. (2013) describes a generative process in which the hypergraph


is grown from a seed hypergraph. This allows control over the degree distribution

through the mechanism in which new nodes are added to the hypergraph. Addition-

ally, whilst the latent space framework has not been considered for the representation

in Figure 4.1.1a, Friel et al. (2016) have developed a latent space model for temporally

evolving bipartite graphs. In this work the authors examine company directors and

boards they associate with.

There has also been a recent interest in edge-exchangeable graph models which,

unlike node-exchangeable graph models, are able to express sparse graphs (see Caron

and Fox (2017), Dempsey et al. (2019), Crane and Dempsey (2018), Campbell et al.

(2018) and Cai et al. (2016)). Many of these models are able to express bipartite

graphs, which can be used to represent a hypergraph (see Figure 4.1.1d). For ex-

ample, Caron and Fox (2017) present an edge-exchangeable model for network data

and, in Section 3.4, they consider how their framework may be applied to bipar-

tite networks. Related to this is the approach of Dempsey et al. (2019) who model

structured interaction processes with an edge-exchangeable framework. This work

considers interactions between sets of nodes, which includes hypergraphs.

Finally, in the probability literature, several authors have studied properties of

random hypergraphs (see Cooley et al. (2018), Elie de Panafieu (2015) and Karonski

and Luczak (2002)) including phase transitions. A particular class of hypergraphs,

known as simplicial hypergraphs, are considered in Kahle (2017). In a simplicial

hypergraph the presence of a hyperedge indicates the presence of all subsets of the

hyperedge. Simplicial hypergraphs can be seen as a special case of a more general

construction termed a simplicial complex, which appear more broadly in the statistics

literature. For example, they are used within topological data analysis (see Salnikov

et al. (2018)) and to determine distances between distributions (see Pronzato et al.

(2018) and Pronzato et al. (2019)). Additionally, they are used in Lunagomez et al.

(2017) to propose simplicial priors for graphical models. Although our work is related

to Lunagomez et al. (2017), we focus on modelling non-simplicial hypergraph data.


The main contributions of this chapter are as follows. First, using the represen-

tation shown in Figure 4.1.1a, we develop a latent space model for non-simplicial

hypergraph data by extending the distance model of Hoff et al. (2002) to the hy-

pergraph setting. We present a specific instance of our model, and comment that

alternative modelling choices can be explored in future work. Second, our model

avoids the computationally expensive full conditional implied by the construction of

Hoff et al. (2002) by relying on tools from computational geometry (see Edelsbrunner

and Harer (2010)). Furthermore, by representing the nodes of the hypergraph in a

low-dimensional latent space, we develop a parsimonious model that is able to express

complex data structures. Third, we draw upon the previously exploited connection

between latent space network models and shape theory to remove non-identifiability

present in our model. More specifically, we infer the latent representation on the

space of Bookstein coordinates which have so far not been explored in this context.

Fourth, by simulating new nodes from the latent representation, we demonstrate how

our model facilitates exploration of the predictive distributions. We also comment

that the latent representation provides a convenient visualisation of the hypergraph

in which more centrally placed nodes have a larger degree. Fifth, we investigate the

theoretical properties of our model and, in particular, we present a framework for

examining the degree distribution. Whilst this proves challenging for our model, our

discussion provides an outline which can be explored for other modelling choices.

The rest of this chapter is organised as follows. In Section 4.2 we provide the

necessary background to introduce our hypergraph model in Section 4.3. Then, in

Section 4.4 we discuss identifiability of our hypergraph model and, in Section 4.5,

we describe our procedure for obtaining posterior samples. The simulation studies

and real data examples are presented in Section 4.7 and 4.8, respectively. Finally, we

conclude with a discussion in Section 4.9.


4.2 Background

In this section we will review the latent space network modelling literature that is

relevant to the model for hypergraph data we introduce in Section 4.3. First, in Section

4.2.1, we discuss the framework introduced in Hoff et al. (2002), where connection

probabilities between pairs of nodes are modelled as a function of a low-dimensional

latent space. Then, in Section 4.2.2, we discuss random geometric graphs (RGGs),

where the presence of an edge between a pair of nodes is determined by the intersection

of convex sets. Finally, in Section 4.2.3 we demonstrate how RGGs can be extended

from the pairwise case to model a restricted class of hypergraphs.

4.2.1 Latent Space Network Modelling

Latent space models were introduced for network data in Hoff et al. (2002) and,

since their introduction, have given rise to a rich modelling literature. The key as-

sumption of this framework is that the nodes of a network can be represented in a

low-dimensional latent space, and that the probability of an edge forming between

each pair of nodes can be modelled as a function of their corresponding latent coor-

dinates. Furthermore, conditional on the iid latent coordinates, the edge between a

given pair of nodes is modelled as independent of all other edges. The dependence

in the network is captured by the latent representation, and this can be made clear

through marginalising over the latent space.

We will first describe a general latent space modelling framework for a network

with N nodes. Let Y = {yij}i,j=1,2,...,N denote the observed (N×N) adjacency matrix,

where yij represents the connection between nodes i and j. For a binary network, we

have that yij = 1 if i and j share an edge and yij = 0 otherwise. Additionally,

we let ui ∈ Rd represent the d-dimensional latent coordinate of the ith node, for


i = 1, 2, . . . , N . The presence of an edge is then given by the following model.

Yij ∼ Bernoulli(pij),

pij = P (yij = 1|ui, uj, θ) =1

1 + exp{−f(ui, uj, θ)},

(4.2.1)

where θ represents additional model parameters and pij denotes the probability of

an edge forming between nodes i and j. The connection probability depends on a

function f that is monotonically decreasing in a measure of similarity between ui and

uj. As an example, the distance model introduced in Hoff et al. (2002) is obtained by

choosing

f(ui, uj, θ) = α− ‖ui − uj‖, (4.2.2)

where ‖ · ‖ is the Euclidean distance, and θ = α represents the base-rate tendency

for edges to form. The function f may also incorporate covariate information so that

nodes which share certain characteristics are more likely to be connected.

We note that the choice of similarity measure will impose characteristics on graphs

generated under this model. If the similarity measure is chosen to be a metric, for

example the Euclidean distance, we know that it satisfies the triangle inequality.

If connections exist between the pairs {i, j} and {i, k}, we know that their latent

coordinates are close in terms of the similarity measure. The triangle inequality

suggests that the node pair {i, k} is also likely to be connected, and so transitive

relationships are likely.

Both asymmetric and symmetric adjacency matrices can be represented in this

framework. Suppose the connections are symmetric and that there are no self ties, so

that we have yij = yji and yii = 0 for i, j = 1, 2, . . . , N . In this case, the likelihood,

conditional on U and θ, is given by

L(U , θ;Y ) ∝∏i<j

P (yij = 1|ui, uj, θ)yij [1− P (yij = 1|ui, uj, θ)]1−yij , (4.2.3)

where U is the (N × d) matrix of latent coordinates such that the ith row of U

corresponds to ui.


The model specified in (4.2.1) and (4.2.3) can be modified in many different ways.

For example, we can model the connection probabilities using the probit link function

instead of the logit link function. Properties of modifications of this type are discussed

in Rastelli et al. (2016). We may also model non-binary connections such as integer

weighted edges. For example, a Poisson likelihood allows us to model edges which

represent the number of interactions between a pair of nodes. Note that this will

require specification of a rate parameter which has a different interpretation to pij.

4.2.2 Random Geometric Graphs

In Section 4.2.1 we outlined the latent space modelling approach for network data.

This framework specifies the probability of an edge forming between a pair of nodes as

a function of their latent coordinates. In this section we will discuss random geometric

graphs (RGGs) which instead model the occurrence of edges as a deterministic func-

tion of the latent coordinates. RGGs can be viewed as a special case of the latent space

framework where, conditional on the latent representation, there is no uncertainty in

the connections. For an in-depth discussion of RGGs see Penrose (2003).

As in the previous section, we will assume that the ith node can be represented

by ui = (ui1, ui2, . . . , uid) ∈ Rd. The presence of an edge {i, j} is modelled through

the intersection of convex sets that are parameterised by ui and uj. There are many

choices of convex sets, and to generate an RGG we choose the closed ball in Rd with

centre ui and radius r. This set is represented Br(ui) = {u ∈ Rd| ‖u − ui‖ ≤ r} ={u ∈ Rd|

√∑dj=1(uj − uij)2 ≤ r

}, and a graph is constructed by connecting each

pair of nodes {i, j} for which Br(ui) ∩Br(uj) 6= ∅. Generating a graph in this way is

equivalent to connecting pairs of nodes for which ‖ui − uj‖ ≤ 2r. An example of this

construction is given in the left and middle panel of Figure 4.2.1.

We now express the likelihood for this model as a function of the latent coordinates,

keeping the notation from Section 4.2.1. The likelihood conditional on U and r is


given by

L(U , r;Y ) ∝∏i<j

1(‖ui − uj‖ ≤ 2r)yij [1− 1(‖ui − uj‖ ≤ 2r)]1−yij . (4.2.4)

By comparing (4.2.3) and (4.2.4), we see that a RGG can be viewed as a latent space

network model where the probability of a connection is expressed as a step function.

More specifically, we have P (yij = 1|ui, uj, θ) = 1(‖ui − uj‖ ≤ 2r) where θ = r. It is

clear from this that, conditional on a set of latent coordinates, a RGG is deterministic.

Note that (4.2.4) is equal to 1 if there is a perfect correspondence between the observed

connections Y and the connections induced by the latent coordinatesU and the radius

r. If there are any connections which do not correspond to each other, then (4.2.4) is

equal to 0.

To specify a more general construction, we define Ai to be a convex set for which

ui ∈ Ai. In the construction above, we let Ai = Br(ui), for i = 1, 2, . . . , N . This

choice of convex set is convenient since, given the radius r and coordinates U , we

are able to generate the graph by considering the distance between pairs of latent

coordinates. For this framework to be computationally appealing, it is important that

the sets Ai are easy to parameterise and their intersections are efficient to compute.

An alternative choice for Ai that is common in the literature is the Voronoi cell, where

Ai ={x ∈ Rd|‖x− ui‖ ≤ ‖x− uj‖, ∀ uj st i 6= j

}(see Section 3.3 of Edelsbrunner

and Harer (2010)).

4.2.3 Random Geometric Hypergraphs

The graph generating procedure described in Section 4.2.2 assumed that edges occur

between pairs of nodes. We can generalise this framework to model hypergraphs

by considering the full intersection pattern of convex sets, and we refer to these

hypergraphs as random geometric hypergraphs (RGHs). In order to do this, we

introduce the concept of a nerve (Section 3.2 of Edelsbrunner and Harer (2010)).

This represents the set of indices for which their corresponding convex regions have a


non-empty intersection and it is given in Definition 4.2.1.

Definition 4.2.1. (Nerve) Let A = {Ai}Ni=1 represent a collection of non-empty con-

vex sets. The nerve of A is given by

Nrv(A) =

{σ ⊆ {1, 2, . . . , N}| ∩

j∈σAj 6= ∅

}. (4.2.5)

Note that the sets {1}, {2}, . . . , {N} are included in Nrv(A) and that |σ| ≤ N for

σ ∈ Nrv(A), where |σ| is the order, or dimension, of the set.

It is clear that the nerve defines a hypergraph where σ ∈ Nrv(A) denotes a

hyperedge. Consider the sets σ1 ∈ Nrv(A) and σ2 ⊂ σ1. It follows immediately that

σ2 ∈ Nrv(A), and all hypergraphs generated by a nerve must have this property.

Hypergraphs of this type are termed simplicial. Kahle (2017) overview properties of

simplicial random hypergraphs along with more general constructions.

In Section 4.2.2, we considered the choice Ai = Br(ui) for generating a RGG. The

nerve for this choice of A is well studied and it is referred to as the Cech complex (see

Section 3.2 of Edelsbrunner and Harer (2010)), as given in Definition 4.2.2.

Definition 4.2.2. (Cech Complex) For a set of coordinates U = {ui}Ni=1 and a radius

r, the Cech complex Cr is given by

Cr = Nrv({Br(ui)}Ni=1

). (4.2.6)

For this framework to be computationally appealing, it is important that the sets

Ai are easy to parameterise and their intersections are efficient to compute. Other well

studied examples of complexes are the Delaunay triangulation (Delaunay (1934)) and

the Alpha complex (Edelsbrunner et al. (1983)) where Ai is specified as the Voronoi

cell for ui (see Section 4.2.2) as the intersection of the Voronoi cell for ui and Br(ui),

respectively.

We now introduce a subset of the Cech complex known as the k-skeleton, which

is given in Definition 4.2.3. This will be revisited in Section 4.3.2.


●

● ●

●

●

●

●

−0.6 −0.4 −0.2 0.0 0.2

0.0

0.2

0.4

0.6

ui1

u i2

●

● ●

●

●

●

●

1

2 3

4

56

7

(a)

●

● ●

●

●

●

●

−0.6 −0.4 −0.2 0.0 0.2

0.0

0.2

0.4

0.6

ui1

u i2

●

● ●

●

●

●

●

1

2 3

4

56

7

(b)

●

● ●

●

●

●

●

−0.6 −0.4 −0.2 0.0 0.2

0.0

0.2

0.4

0.6

ui1

u i2

●

● ●

●

●

●

●

●

● ●

●

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●

1

2 3

4

56

7

(c)

Figure 4.2.1: Example of a Cech complex. Left: Br(ui) for {ui = (ui1, ui2)}7i=1 in R2.

Middle: the graph obtained by taking pairwise intersections. Right: the hypergraph

obtained by taking intersections of arbitrary order. The shaded region between nodes

3, 5 and 6 indicates a hyperedge of order 3.

Definition 4.2.3. (k-skeleton of the Cech complex) Let Cr denote the Cech complex,

as given in Definition 4.2.2. The k-skeleton of Cr is given by

C(k)r = {σ ∈ Cr||σ| ≤ k}. (4.2.7)

C(k)r represents the collection of sets in Cr which are of order that is less than or

equal to k. Note that the k-skeleton can also be defined more generally for any nerve.

Example 4.2.1: Figure 4.2.1 depicts an example of a Cech complex, where Cr =

{{1}, {2}, {3}, {4}, {5}, {6}, {7}, {2, 4}, {2, 7}, {3, 5}, {3, 6}, {5, 6}, {3, 5, 6}}. The k-

skeletons are given by C(1)r = {{1}, {2}, {3}, {4}, {5}, {6}, {7}}, C(2)r = C(1)r ∪ {{2, 4},

{2, 7}, {3, 5}, {3, 6}, {5, 6}} and C(3)r = C(2)r ∪ {3, 5, 6}.

4.3 Latent space hypergraphs

In this section we will introduce a model for hypergraph data which builds upon the

models discussed in Section 4.2. Our model will balance the computational aspects of


latent space network modelling (Section 4.2.1) with the approach of RGGs (Section

4.2.2) and RGHs (Section 4.2.3). The aims of our modelling framework are given

in Section 4.3.1, notation and set-up are given in Section 4.3.2, and the generative

model and likelihood are given in Section 4.3.3. Finally, in Section 4.3.4 we extend

the model to a more flexible scenario.

4.3.1 Motivation

Consider a co-authorship network where nodes represent authors and edges indicate

which authors contributed to a given paper. In this context, papers that have been

written by more than two authors are naturally represented by a hyperedge. The

hypergraph model discussed in Section 4.2.3 is likely not appropriate for these data

since a set of authors having worked on a paper does not imply that all subsets of

those authors have also written papers together. This motivates us to develop a model

for non-simplicial hypergraphs.

In this section we will build upon the pairwise graph and hypergraph models

introduced in Section 4.2. We introduce a model for hypergraph data which represents

the nodes of the network in a low-dimensional space and, unlike the approach in

Section 4.2.3, is able to express a broad class of hypergraphs. The model will be

developed with the following objectives.

1. Convenient likelihood

The model of Hoff et al. (2002) specifies the connection probabilities for all node

pairs and so the likelihood, conditional on the latent coordinates, for this model

is a function of a 2D tensor. In the hypergraph analogue of this framework, the

conditional likelihood would be a function of a kD tensor, where k is the order

of the hyperedges. Evaluation over a kD tensor is an order O(Nk) computation,

which becomes increasingly computationally expensive as the number of nodes

N and the hyperedge order k grow. In contrast to this, graphs and hypergraphs

generated in the RGG (Section 4.2.2) and RGH (Section 4.2.3) framework are


a deterministic function of the latent space. RGHs can be computed efficiently

without considering all possible hyperedges, however the conditional likelihood

is equal to either 0 or 1 and this may hinder model fitting. Our aim is to develop

a model which draws on the advantages of each of these approaches. We present

a model with a likelihood that is amenable to Bayesian computation and whose

evaluation does not rely on a calculation over a tensor.

2. Simple to describe support

Since the edges in the model of Hoff et al. (2002) exhibit uncertainty after con-

ditioning on the latent coordinates, it is clear that the model can express the

entire space of graphs on N nodes with pairwise interactions. In the RGG

framework (Section 4.2.2), connections are instead modelled deterministically

through the intersection of convex sets. It is not clear in general how to charac-

terise the space of graphs that represent the support for this generative model.

Furthermore, when this framework is extended to model non-simplicial hyper-

graphs, the support for the generative model is complicated further. Based on

the approach of RGHs (Section 4.2.3), we develop a model for non-simplicial

hypergraph data whose support is straightforward to describe.

Throughout this section we will describe our model for a hypergraph on N nodes

with a maximal hyperedge order K, where 2 ≤ K ≤ N . We let e ⊆ {1, 2, . . . , N}

denote a hyperedge and the presence of e is denoted by ye = 1. Otherwise, we let

ye = 0. Where necessary, we will denote a hyperedge of order k by ek so that |ek| = k.

4.3.2 Combining k-skeletons

To extend the model in Section 4.2.3 to express non-simplicial hypergraphs, we in-

troduce additional radii. As an example, consider two radii which we denote by r2

and r3. For the same set of latent coordinates U , each of these radii will give rise to

Cech complex which we denote by Cr2 and Cr3 , respectively. By varying r2 and r3 we


●

● ●

●

●

●

●

−0.6 −0.4 −0.2 0.0 0.2

0.0

0.2

0.4

0.6

ui1

u i2

●

● ●

●

●

●

●

1

2 3

4

56

7

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● ●

●

●

●

●

−0.6 −0.4 −0.2 0.0 0.2

0.0

0.2

0.4

0.6

ui1

u i2

●

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●

●

●

●

1

2 3

4

56

7

●

● ●

●

●

●

●

−0.6 −0.4 −0.2 0.0 0.2

0.0

0.2

0.4

0.6

ui1

u i2

●

● ●

●

●

●

●

●

● ●

●

●

●

●

1

2 3

4

56

7

Figure 4.3.1: Example of a Cech complex. Left: Br2(u) for each of 6 points in R2.

Middle: Br3(u) for each of 6 points in R2. Right: ∪3k=2D

(k)rk .

are able to control the edges that are present in each of Cr2 and Cr3 . Now suppose

for each of these complexes we only consider hyperedges of a specific order. We can,

for example, construct a hypergraph by taking the union of the order 2 hyperedges

in Cr2 and the order 3 hyperedges in Cr3 . This construction removes the simplicial

constraint on the hypergraphs, and an example of this is given in Figure 4.3.1. We

refer to hypergraphs constructed in this way as non-simplicial random geometric hy-

pergraphs (nsRGH), and Definition 4.3.1 details this construction for a hypergraph

on N nodes with maximal hyperedge order K.

Definition 4.3.1. (non-simplicial RGH) Consider r = (r2, r3, . . . , rK) which satisfy

rk > rk−1 for k = 3, 4, . . . , K. We define the non-simplcial RGH (nsRGH) on N

nodes as the hypergraph with hyperedges given by ∪Kk=2D(k)rk , where D(k)

rk = C(k)rk \ C(k−1)rk

denotes the hyperedges of exactly order k in the Cech complex with radius rk, and C(k)rk

is as in Definition 4.2.3.

Example 4.3.1: Consider the non-simplicial hypergraph shown in the right panel of

Figure 4.3.1. We have D(2)r2 = {{2, 4}, {3, 5}, {5, 6}} and D(3)

r3 = {3, 5, 6}.

In Definition 4.3.1, constraints are imposed on the radii r = (r2, r3, . . . , rK) to

ensure that the generated hypergraphs are non-simplicial. For example, if r3 < r2

and the hyperedge {i, j, k} is present in the hypergraph, then it follows that the

hyperedges {i, j}, {i, k} and {j, k} must also be in the hypergraph.


We comment here that the Cech complex is also used in Lunagomez et al. (2017)

to propose simplicial priors for graphical models, however our setting differs from this

since we focus on modelling non-simplicial hypergraph data.

4.3.3 Generative Model and Likelihood

The procedure of generating non-simplicial hypergraphs given in Definition 4.3.1 is

deterministic conditional on U . Therefore, similarly to the RGG in Section 4.2.2,

the likelihood of an observed hypergraphs will be one only when there is a perfect

correspondence between the observed and generated hyperedges. Furthermore, it is

not straightforward to characterise the space of hypergraphs that can be expressed

via the nsRGH procedure. We address these issues in this section by considering a

modification of Definition 4.3.1.

Let GN,K denote the space of hypergraphs on N nodes with maximum hyperedge

order K. We write GN,K = (VN , EN,K), where VN = {1, 2, . . . , N} denotes the node

labels and EN,K denotes the set of possible hyperedges up to order K on N nodes.

Let EN,k represent the possible hyperedges of exactly order k on N nodes so that

EN,K = ∪Kk=2EN,k. Let ϕk ∈ [0, 1] denote the probability of modifying the state of a

hyperedge of order k, for k = 2, 3, . . . , K. Then, for ek ∈ EN,k, we sample a variable

Sk ∼ Bernoulli(ϕk) and let

y(g∗)

ek=

0, if sk = 1 and y

(g)ek = 1

1, if sk = 1 and y(g)ek = 0

yek , if sk = 0

(4.3.1)

where gN,K(U , r) denotes the nsRGH induced fromU and r as in Definition 4.3.1, and

ye(g)k

denotes the state of the hyperedge ek in gN,K(U , r). We let g∗N,K(U , r) denote

the hypergraph sampled from our model. From (4.3.1), we see that ϕk controls the

amount of modification of the hyperedges of order k.

The final aspect of the generative model is to assign a probability distribution on


Algorithm 7 Sample a hypergraph g∗N,K given N,K, r,ϕ, µ and Σ.

Sample U = {ui}Ni=1 such that uiiid∼ N (µ,Σ), for i = 1, 2, . . . , N .

For k = 2, 3, . . . , K,

a) Given U and rk, check which ek = {i1, ı2, . . . , ik} ∈ EN,k satisfy y(g)ek = 1.

To determine if y(g)ek = 1, check that ∩kl=1Brk(uil) 6= ∅.

b) For all ek ∈ EN,k, sample Sk ∼ Bernoulli(ϕk).

Let y(g∗)ek =

(y(g)ek + sk

)mod 2

the latent coordinates U , and we assume that uiiid∼ N (µ,Σ), for i = 1, 2, . . . , N . This

reflects the intuition that nodes placed more centrally in the latent representation

are likely to share the greatest number of connections. A hypergraph can now be

generated by the procedure given in Algorithm 7.

We can express the likelihood of an observed hypergraph hN,K ∈ GN,K , conditional

on U , r and ϕ, by considering the discrepancy between the hyperedge configurations

in hN,K and gN,K(U , r). For k = 2, 3, . . . , K, let

dk(gN,K(U , r), hN,K) =∑

ek∈EN,k

|y(g)ek− y(h)ek

| (4.3.2)

denote the distance between the order k hyperedges in gN,K(U , r) and hN,K , where y(g)ek

and y(h)ek represent an order k hyperedge in gN,K(U , r) and hN,K , respectively. Note

that no modifications have been made to the hypergraph gN,K(U , r). The measure of

distance (4.3.2) corresponds to the number of hyperedges which differ between hN,K

and gN,K(U , r). This is equivalent to the Hamming distance and is related related to

the l1 norm and the exclusive or (XOR) operator. Evaluating this distance does not

require the∑K

k=2

(Nk

)computations suggested by (4.3.2), and we can instead evaluate

the discrepancy by only considering hyperedges that are present in gN,K(U , r) and

hN,K . In practice this is likely to be far less than the number of possible hyperedges

and details of this calculation are discussed in Appendix B.5.2.

Given this notion of hypergraph distance the likelihood of observing hN,K , condi-


tional on U , r and ϕ, can be written as

L(U , r,ϕ;hN,K) ∝K∏k=2

ϕdk(gN,K(U ,r),hN,K)k (1−ϕk)

(Nk)−dk(gN,K(U ,r),hN,K). (4.3.3)

We obtain (4.3.3) by considering which hyperedges in gN,K(U , r) must have their

state modified to match the hyperedges in hN,K , and which hyperedges are the same

as in hN,K . For order k hyperedges which differ, the probability of switching the

hyperedge state is given by ϕk. Since our likelihood is of the same form as Lunagomez

et al. (2019, proof of Proposition 3.1), it follows that hypergraphs with a greater

number of hyperedge modification are less likely for 0 < ϕk < 1/2 and so (4.3.3)

behaves in an intuitive way.

The model specification is complete with the following priors for k = 2, 3, . . . , K.

µ ∼ N (mµ,Σµ) Σµ ∼ W−1(Φ, ν), rk ∼ exp(λk), and ϕk ∼ Beta(ak, bk),

(4.3.4)

where the priors in (4.3.4) are chosen for computational convenience.

4.3.4 Extensions

In Algorithm 7 the number of order k edges is controlled by varying the parameter

rk. However, the constraint rk+1 > rk, for k = 2, 4, . . . , K − 1, implies rk will impact

the higher order hyperedges. This may limit the types of hypergraphs that can be

expressed.

We improve the model flexibility by introducing an additional modification param-

eter for each hyperedge order. In Algorithm 7 the noise ϕk is applied independently

across all hyperedges of order k. Alternatively, we can modify each hyperedge depend-

ing on its state in gN,K(U , r). For k = 2, 3, . . . , K, let ψ(0) = (ψ(0)2 , ψ

(0)3 , . . . , ψ

(0)K ) ∈

[0, 1] denote the probability of modifying the state of a hyperedge in gN,K(U , r) from

absent to present, and let ψ(1) = (ψ(1)2 , ψ

(1)3 , . . . , ψ

(1)K ) ∈ [0, 1] denote the probability

of modifying the state of a hyperedge in gN,K(U , r) from present to absent. Sup-

pose our observed hypergraph suggests that there are many hyperedges of order 2


Algorithm 8 Sample a hypergraph g∗N,K given N,K, r,ψ(0),ψ(1), µ and Σ.

Sample U = {ui}Ni=1 such that uiiid∼ N (µ,Σ), for i = 1, 2, . . . , N .

For k = 2, 3, . . . , K,

a) Given U and rk, check which ek = {i1, ı2, . . . , ik} ∈ EN,k satisfy yek = 1.

To determine if yek = 1, that ∩kl=1Brk(uil) 6= ∅.

b) For all ek ∈ EN,k

If y(g)ek = 1, set y

(g∗)ek = 0 with probability ψ

(1)k .

If y(g)ek = 0, set y

(g∗)ek = 1 with probability ψ

(0)k .

and few hyperedges of order 3. By increasing the modification noise ψ(1)3 we can con-

trol additional hyperedges that may appear in the hypergraph due to the constraint

r3 > r2. This generative model is summarised in Algorithm 8, and as commented

in Section 4.3.3, Algorithm 8 can be implemented without the suggested∑K

k=2

(Nk

)computations. Further details on the procedure for checking which hyperedges are

present in gN,K(U , r) are discussed in Appendix B.5.1 with details of the hyperedge

modifications given in Appendix B.2.

As in Section 4.3.3, the likelihood of observing hN,K is based on a distance metric,

d(ab)k (gN,K(U , r), hN,K) = #{ek ∈ EN,k|y(g)ek

= a ∩ y(h)ek= b}, (4.3.5)

which records the number of hyperedges that have state a ∈ {0, 1} in gN,K(U , r) and

state b ∈ {0, 1} in hN,K . For example, d(01)k (gN,K(U , r), hN,K) represents the number

of hyperedges absent in gN,K(U , r) and present in hN,K . Efficient evaluation of (4.3.5)

is discussed in Appendix (B.5.2) and the likelihood conditional on U , r,ψ(1) and ψ(0)

is given by

L(U , r,ψ(1),ψ(0);hN,K

)∝

K∏k=2

[(ψ

(1)k

)d(10)k (gN,K(U ,r),hN,K) (1− ψ(1)

k

)d(11)k (gN,K(U ,r),hN,K)

×(ψ

(0)k

)d(01)k (gN,K(U ,r),hN,K) (1− ψ(0)

k

)d(00)k (gN,K(U ,r),hN,K)].

(4.3.6)


We obtain (4.3.6) in a similar way to (4.3.3), where we distinguish between hy-

peredges that are present and absent in the induced hypergraph. Note that (4.3.6) is

equivalent to (4.3.3) when ψ(1)k = ψ

(0)k , for k = 2, 3, . . . , K.

The model specification is complete with the following prior distributions, for

k = 2, 3, . . . , K,

µ ∼ N (mµ,Σµ), Σ ∼ W−1(Φ, ν), rk ∼ exp(λk),

ψ(0)k ∼ Beta

(a(0)k , b

(0)k

), and ψ

(1)k ∼ Beta

(a(1)k , b

(1)k

). (4.3.7)

where priors in (4.3.7) are chosen for computational convenience.

4.4 Identifiability

Let gN,K(U , r) denote the nsRGH obtained from U and r, and let g∗N,K be the hy-

pergraph obtained by modifying the hyperedges in gN,K(U , r) according to ϕ (see

Algorithm 7). By conditioning on gN,K(U , r), we can decompose the conditional

distribution for g∗N,K in the following way.

p(g∗N,K , gN,K(U , r)|µ,Σ,ϕ, r) = p(g∗N,K |gN,K(U , r),ϕ)p(gN,K(U , r)|µ,Σ, r). (4.4.1)

An equivalent decomposition for the model outlined in Algorithm 8 can also be ex-

pressed.

The probability of occurrence of a hyperedge in gN,K(U , r) is a function of the

distances between the latent coordinates U . Therefore the conditional distribution

p(gN,K(U , r)|µ,Σ, r) is invariant to distance-preserving transformations of U . Addi-

tionally, we observe that scaling U and r by the same factor results in a source of

model non-identifiability.

To remove these sources of non-identifiability, we define U on the Bookstein space

of coordinates (see Bookstein (1986) and Section 2.3.3 of Dryden and Mardia (1998)).

Bookstein coordinates define a translation, rotation and re-scaling of the points U

with respect to a set of anchor points and, since these anchor points remain fixed


throughout model fitting, the radii r are also appropriately re-scaled. For details of

the Bookstein transformation see Appendix B.1.

In the latent space network modelling literature, it is typical to use Procrustes

analysis (Section 5 of Dryden and Mardia (1998)) as a post-processing step to remove

the effect of distance preserving transformations of U . Due to the non-identifiability

associated with scaling U and r, we note that this approach is not sufficient for

removing all sources of non-identifiability in our model.

From (4.4.1), we see that hyperedges can either arise from gN,K(U , r) or the hyper-

edge modification. To maintain the properties imposed on the hypergraph from the

construction of gN,K(U , r), we wish to keep the parameters ϕ relatively small. How-

ever, when generating sparse hypergraphs from our model, it will become increasingly

difficult to distinguish between these competing sources of hyperedges. Therefore we

will observe model non-identifiability in the sparse regime.

4.5 Posterior Sampling

To sample from the models specified in Sections 4.3.3 and 4.3.4 we implement an

MCMC scheme. In Section 4.5.1 we provide a high-level description of the posterior

sampling procedure for the model detailed in Algorithm 8 and comment that this can

easily be modified to the model detailed in Algorithm 7. Where appropriate we refer

the reader to the relevant sections of the appendix for more detail.

4.5.1 MCMC scheme

To obtain posterior samples from the model presented in Algorithm 8 we implement

a Metropolis-Hastings-within-Gibbs MCMC scheme (see Section 6.4.2 of Gamerman

and Lopes (2006)). We update the latent coordinates U and radii r with a Metropolis

Hastings (MH) step, and the remaining parameters are updated via Gibbs samplers.

The priors for the model are specified in (4.3.7).


When updating the latent coordinates we use a random walk MH. As discussed in

Section 4.4, we define U on the Bookstein space of coordinates and so a set of anchor

points will remain fixed throughout the MCMC scheme. For ui ∈ Rd, let the anchor

points be denoted by u1 and u2. For i = 3, 4, . . . , N , we propose u∗i = ui + εu where

εu ∼ N (0, σuId), and for i = 1, 2 we let u∗i = ui. We then accept U ∗ = {u∗i }Ni=1 as a

sample from p(U |µ,Σ, r,ψ(0),ψ(1), hN,K) with probability

min

{1,L(U ∗, r,ψ(1),ψ(0);hN,K)p(U ∗|µ,Σ)

L(U , r,ψ(1),ψ(0);hN,K)p(U |µ,Σ)

}, (4.5.1)

where p(U |µ,Σ) =∏N

i=1 p(ui|µ,Σ). Since the proposal mechanism is symmetric in

terms of U and U ∗, the term associated with this does not appear in (4.5.1).

The acceptance ratio (4.5.1) is for the entire (N × d) latent representation U . As

N grows, jointly proposing all latent coordinates will become increasingly inefficient

due to the dimension of U . Alternatively we can partition the latent coordinates into

disjoint sets {Ul}Ll=1 such that ∪Ll=1Ul = {1, 2, . . . , N}, and perform the MH update

for each of these L sets separately. This approach will be used in the examples in

Sections 4.7 and 4.8, and details of this are given in Algorithm 9.

To update r, we let r∗ = (r∗2, r∗3, . . . , r

∗K) where r∗k = rk + εr and εr ∼ N (0, σr).

Then, if r∗k+1 > r∗k for k = 2, 3, . . . , K − 1, we accept r∗ as a sample from p(r|U ,ψ(0),

ψ(1), µ,Σ) with probability

min

{1,L(U , r∗,ψ(1),ψ(0);hN,K)p(r∗|λ)

L(U , r,ψ(1),ψ(0);hN,K)p(r|λ)

}, (4.5.2)

where p(r|λ) =∏K

k=2 p(rk|λk). Otherwise, we reject r∗.

All other parameters can be sampled directly from their full conditionals, and the

details of the MCMC scheme for imax iterations are given in Algorithm 9. Initialisation

for the MCMC is non-trivial, and a discussion of this can be found in Appendix B.4.

To implement the MCMC scheme, there are a number of computational considera-

tions we must address (see Appendix B.5). Firstly, to evaluate the likelihood we need

to determine the hyperedges in the hypergraph generated from U and r, g(U , r).

We rely on methodology from the computational topology literature to do this, and


Algorithm 9 MCMC scheme to obtain posterior samples of U , r,ψ(0),ψ(1),Σ and

µ.

Specify N,K,L, imax ∈ N.

Initialise

Determine initial values for U , r,ψ(0),ψ(1),Σ and µ using Algorithm 12 (see

Appendix B.4).

For it in 1, 2, . . . , imax

1) Sample µ(i) from p(µ|U ,Σ,mµ,Σµ) (see Appendix B.3.1).

2) Sample Σ(i) from p(Σ|U , µ,Φ, ν) (see Appendix B.3.2).

3) Partition {u3, u4, . . . , uN} into L sets Ul.

For l = 1, 2, . . . , L

For i ∈ Ul, propose u∗i = ui + εu, where εu ∼ N (0, σuId)

Accept proposal with probability (4.5.1).

4) For k = 2, 3, . . . , K, propose r∗, where rk∗ = rk + εr and εr ∼ N (0, σr)

Accept proposal r∗ with probability (4.5.2).

5) For k = 2, 3, . . . , K

Sample ψ(0)k from p(ψ

(0)k |U , r, hN,K , a

(0)k , b

(0)k

)(see Appendix B.3.3).

Sample ψ(1)k from p

(ψ

(1)k |U , r, hN,K , a

(1)k , b

(1)k

)(see Appendix B.3.4).

details of the approach used are given in Appendix B.5.1. Secondly, to calculate the

likelihood we note that we can avoid the summation over the entire set of hyperedges

as suggested by (4.3.5) and, in Appendix B.5.2, we discuss this in more detail.

4.6 Theoretical Results

In this section we study the behaviour of the node degree in the hypergraph model

detailed in Algorithm 7. We begin by making observations on the structure of our

model in Section 4.6.1, and then consider the probability of a hyperedge occurring in

a nsRGH in Section 4.6.2. Finally, we present our results for the degree distribution


in Section 4.6.3. We comment here that it is straightforward to extend the results in

this section to the model detailed in Algorithm 8.

4.6.1 Observations

To study the properties of the node degree we first observe that the nodes in our

hypergraph model are exchangeable, and so we focus on obtaining results for the ith

node. To guide the structure of our proofs we make the following observations.

(O1) A hypergraph generated from our model is a modification of a nsRGH

To generate a hypergraph from our model, we first determine hyperedges through

the intersection of sets Br(ui) for i = 1, 2, . . . , N . The indicators for the presence

and absence of hyperedges are then modified according to the noise parameters

ϕ. Since the modifications are applied independently, we can view our hyper-

graph model as an Erdos-Renyi modification of a nsRGH generated from U and

r.

(O2) Hyperedges of different orders occur independently in our model

Conditional on the latent coordinates U and radii r, the hyperedges of order k

occur independently of hyperedges of order k′ 6= k. Therefore, we can consider

the degree distribution as a sum over the degree distributions for hyperedges of

exactly order k.

Throughout this section, we will assume that the number of nodesN and maximum

hyperedge order K are fixed. We let g(U , r) = g ∈ GN,K denote the nsRGH generated

from the coordinates U and radii r. A hypergraph is generated from our model by

modifying g with noise ϕ, and we denote this hypergraph by g∗ ∈ GN,K . Additionally,

we denote the degree of order k hyperedges and the overall degree of the ith node in

g by Degg(i,k) =∑{ek∈EN,k|i∈ek} y

(g)ek and Degg(i) =

∑Kk=2 Deg(i,k), respectively.

Using this notation, we recall the decomposition for hN,K in (4.4.1) and comment

that this follows from (O1). Finally, we will assume that the covariance matrix for the


latent coordinates Σ is diagonal so that Σll = σ2l for l = 1, 2, . . . , d and Σlm = 0 for

l 6= m. Note that this assumption is not restrictive since for any normally distributed

set of points in Rd we can apply a distance-preserving transformation which maps the

covariance matrix onto a diagonal matrix.

4.6.2 Properties of a nsRGH

In this section we consider the probability an order k hyperedge occurring in a nsRGH

generated from U and r, and we denote this by pek = P (y(g)ek = 1|µ,Σ, rk). We present

results for k = 2 in Section 4.6.2 and discuss the connection probability for k ≥ 3 in

Section 4.6.2.

Connection Probabilities for k = 2

Recall that an edge e2 = {i, j} is present in g(U , r) if Br2(ui) ∩ Br2(uj) 6= ∅. Hence,

to obtain an expression for the occurrence probability pe2 , we consider the probability

of the coordinates ui and uj lying within distance 2r2 of each other. This probability

is given in Proposition 4.6.1 and follows by considering the distribution of a squared

Normal random variable.

Proposition 4.6.1: Let Ui ∼ N (µ,Σ), for i = 1, 2, . . . , N , and Σ = diag(σ21, σ

22, . . . , σ

2d).

The probability of an edge e2 = {i, j} occurring in g(U , r) is given by

pe2 = P (‖Ui − Uj‖ ≤ 2r2|Σ) =

∫ (2r2)2

0

d∑l=1

f

(z;

1

2, 4σ2

l

)dz, (4.6.1)

where f(z; a, b) =ba

Γ(a)za−1e−bz is the pdf of a Γ(a, b) random variable.

Proof. See Appendix B.6.1.

To check the validity of this result, Figure 4.6.1 shows an empirical estimate of

pe2 compared to the result in Proposition 4.6.1 for two choices of Σ. Next, we will

consider the connection probability pek for k ≥ 3.


0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

rk

p ek

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

rk

p ek

(b)

Figure 4.6.1: Estimate of probability of a hyperedge occurring for k = 2 (red, solid),

k = 3 (pink, dot), k = 4 (purple, dash-dot), k = 5 (grey, dash) as a function of rk.

Points were simulated from a Normal distribution with µ = (0, 0), and Σ = I2 in (a)

and Σ =(2 00 1

)in (b), and the theoretical probability for k = 2 is plotted for each

case (black, dash).

Connection Probabilities for k ≥ 3

To determine pek for k ≥ 3, we first note that the edge ek = {i1, i2, . . . , ik} is present

in g(U , r) if ∩kl=1Brk(uil) 6= ∅. This condition is equivalent to the coordinates {uil}kl=1

being contained within a ball of radius rk (see section 3.2 of Edelsbrunner and Harer

(2010)). This is depicted in Figure B.5.1, and for more details of this see Appendix

B.5.1. Given this observation, we can determine pek by finding the probability of

exactly k points falling within a ball of radius rk.

For normally distributed coordinates, the probability of a point falling within a

ball of radius rk and centre c is given by

P (u ∈ Brk(c)|µ,Σ) =

∫Brk (c)

p(u|µ,Σ) du. (4.6.2)

(4.6.2) presents a challenging computation and results for this integral are pro-


vided in Gilliland (1962) when d = 2. It is therefore not possible to obtain an exact

expression for pek when k ≥ 3 and we instead rely on empirical approximations.

Figure 4.6.1 shows Monte Carlo estimates of pek for increasing rk, where the case

k = 2 is provided for reference. Points were sampled from N (0,Σ) and the left and

right panel show the estimated connection probabilities for Σ = I2 and Σ =(2 00 1

),

respectively. From the figure, we see that a larger radius is required to obtain the

same probability of connection as k grows. Additionally, by comparing the left and

right panels of Figure 4.6.1, we see that, as the elements of Σ increase, the radii rk

must also increase to obtain the same probability of a connection.

4.6.3 Degree Distribution for the ith Node

In this section we investigate the degree distribution for the ith node. This is presented

in Theorem 4.6.1 for hyperedges of order k = 2 and k = 3 and, from this result,

we obtain Lemma 4.6.1 which describes the expected degree of the ith node. In

Theorem 4.6.1 we present an exact expression for the degree distribution of order

k = 2 hyperedges, and rely on an approximation for k = 3 hyperedges. We comment

on the quality of this approximation below.

Lemma 4.6.1. Let pek denote the probability of the hyperedge ek occurring in the

nsRGH g, and let g∗ be the hypergraph obtained by modifying g with noise ϕ. The

expected degree of the ith node is given by

E[Degg

∗

(i)|ϕ,Σ, r]

=K∑k=2

(N − 1

k − 1

)[(1−ϕk)pek + ϕk(1− pek)] . (4.6.3)


Theorem 4.6.1: Let g represent the nsRGH generated from U and r, and let g∗ be

the hypergraph obtained by modifying g with noise ϕ. It follows that


0 5 10 15 20

0.00

0.10

0.20

Degree

Pro

babi

lity

(a) k = 2.

0 10 20 30 40 50

0.00

0.10

0.20

Degree

Pro

babi

lity

(b) k = 3.

Figure 4.6.2: Comparison of theoretical (black, dashed) and simulated (red, solid)

degree distribution for a hypergraph with N = 20 and K = 3. Figures 4.6.2a and

4.6.2b show the degree distribution for hyperedges of order k = 2 and k = 3, respec-

tively. The theoretical degree distribution for k = 3 is calculated using a monte carlo

estimate of pe3 .

1. For k=2:

Degg∗

(i,2)|ϕ,Σ, r ∼ Binomial (N − 1, (1−ϕ2)pe2 + ϕ2(1− pe2)) , (4.6.4)

where pe2 = P (y(g)e2 = 1|Σ, r2) is the probability of e2 being present in g.

2. For k=3:

Degg∗

(i,3)|ϕ,Σ, r ∼ Poisson

∑{e3∈EN,3|i∈e3}

(N − 1

2

)[(1−ϕ3)pe3 + ϕ3(1− pe3)]

,

(4.6.5)

where pe3 = P (y(g)e3 = 1|Σ, r3) is the probability of e3 being present in g and

X ∼ f(x) indicates that X is approximately distributed according to f(x).



Theorem 4.6.1 and the corresponding Lemma describe how the model parameters

affect the degree distribution. The probability of connection pk = (1−ϕk)pek+ϕk(1−

pek) reflects observation (O1), and we note that equivalent degree distributions can

be obtained from different choices for ϕk and pek . For example, small pek and large

ϕk will behave similarly to large pek and small ϕk. However, the characteristics of

the resulting hypergraphs will differ significantly in each of these cases. The Poisson

approximation in Theorem 4.6.1 is only appropriate when p3 is small, and we comment

that improvements may be made to this approximation using methodology presented

in Teerapabolarn (2014). As a sanity check, Figure 4.6.2 compares the theoretical

degree distribution with the simulated degree distribution for a hypergraph with N =

20 and K = 3. We also compare the theoretical and observed degree distributions for

the data example presented in Section 4.8.3.

4.7 Simulations

In this section we describe three different simulation studies. We begin in Section

4.7.1 with an investigation of the flexibility of our modelling approach in comparison

with two other hypergraph models from the literature. Then, in Section 4.7.2, we

examine the predictive degree distribution conditional on an observed hypergraph,

and in Section 4.7.3 we consider the robustness of our model with respect to different

types of misspecification.

4.7.1 Model depth comparisons

In this study we explore the range of hypergraphs that can be expressed in our mod-

elling framework. We compare this with two other statistical models from the lit-

erature, which have been designed according to different modelling aims. For each

model, we specify several cases which are designed to highlight particular aspects of

the model. Then, we simulate hypergraphs for each case and record summary statis-


tics to characterise the simulated hypergraphs. We will begin by outlining the models

of Stasi et al. (2014) and Ng and Murphy (2018) for a hypergraph with N nodes,

and then describe and justify the choice of cases and summary statistics. Finally, we

discuss the results of the simulations.

We first describe an extension of the β-model for random graphs (see Holland and

Leinhardt (1981)), introduced by Stasi et al. (2014). In this model each node in the

hypergraph is assigned a parameter which controls its tendency to form edges, and

we denote this parameter by βi, for i = 1, 2, . . . , N . Let yek = 1 denote the presence

of the hyperedge ek = {i1, i2, . . . , ik} ⊆ {1, 2, . . . , N} for k ≥ 2. The probability of

the hyperedge ek occurring is then given by

p(yi1i2...ik = 1) =exp{βi1 + βi2 + . . . βik}

1 + exp{βi1 + βi2 + . . . βik}. (4.7.1)

Since the hyperedges are assumed to occur independently conditional on β = (β1, β2, . . . ,

βN), the likelihood is obtained by taking the product of Bernoulli likelihoods over all

possible hyperedges EN,K . This likelihood can be shown to belong to the exponential

family. Stasi et al. (2014) introduce several variants of this model, however we only

rely on the above for our study.

Next, we overview the model introduced in Ng and Murphy (2018) which assumes

that hyperedges can be clustered according to their topic and size. In this context, the

topic clustering implies that the hyperedges can be partitioned into latent classes and

the probability of a node belonging to a hyperedge depends on its latent class. As an

example, consider a coauthorship network where papers are represented as hyperedges.

We may classify papers according to their academic discipline and impose that certain

authors are more likely to contribute to papers within different disciplines. The size

clustering is with respect to the hyperedge order, and this allows the model to capture

variation in the size of hyperedges. To specify this model, we assume T topic clusters

and S size clusters. It is assumed that the ith node belongs to a hyperedge with

size label s and topic label t with probability αsφit, so that α = (α1, α2, . . . , αS)

controls the size clusters and φ = {φit}i=1,2,...,N,t=1,2,...,T controls the topic clusters.


Additionally, we let π = (π1, π2, . . . , πT ) and τ = (τ1, τ2, . . . , τS) denote the prior

topic and size assignment probabilities, respectively. To write down the likelihood,

we let xij = 1 indicate that the ith node belongs to the jth hyperedge, z(1)jt = 1 indicate

that the jth hyperedge has topic label t, and z(2)js = 1 indicate that the jth hyperedge

has size label s. The likelihood is then given by

L(x, z(1), z(2); θ) =M∏j=1

T∏t=1

S∏s=1

[πtτs

N∏i=1

(αsφit)xij(1− αsφit)1−xij

]z(1)jt z(2)js. (4.7.2)

Finally, to ensure the model is identifiable, we set αS = 1. Ng and Murphy (2018)

also introduce a version of this model which only assumes a topic clustering, but we

do not use this for our study.

In this simulation study we consider the models detailed above and the hypergraph

model described in Algorithm 8. Before describing the set up, we note that each of

these models has been designed for a different purpose. The β-model of Stasi et al.

(2014) allows fine control over the degree distribution through the parameters β, and

the model of Ng and Murphy (2018) is designed to describe hyperedges which ex-

hibit a clustering structure. The modelling choices in our approach impose different

characteristics on the resulting simulated hypergraphs. By determining the hyper-

edges from the Cech complex, we expect that the resulting hypergraphs will exhibit

transitivity since, if {i, j} and {i, k} lie sufficiently close to be connected, then {j, k}

is likely to also be present in the hypergraph. Additionally, the presence of hyper-

edges {i, j}, {i, k} and {j, k} suggests that the hyperedge {i, j, k} is more likely to be

present. To model hypergraphs with different characteristics within our framework,

we can investigate the effect of changing the assumed distribution of the latent coor-

dinates and using an alternative simplicial complex. This point will be discussed in

more detail in Section 4.9.

We now describe the set of metrics used to capture the above model behaviours.

To capture the connectivity patterns we expect from our model, we record counts

for the subgraphs depicted in Figure 4.7.1a. To measure the degree distribution and


●

●

●●

●

●

(a1) 3 star

●

●

●

●

●

●

●

●

(a2) 4 star

●

●

●●

●

●

(a3) h1

●

●

●●

●

●

(a4) h2

●

●

●●

●

●

(a5) h3

(a) Motifs counted in model simulations.

●

●

●●

●

●

(b1) m1

●

●

●●

●

●

(b2) m2

●

●

●●

●

●

(b3) m3

(b) Additional motifs.

Figure 4.7.1: Depiction of motifs considered in Sections 4.7 and 4.8.

spread of hyperedge orders we record the percentiles of the node degrees and edge

sizes, respectively. Additionally, we record the density of the hyperedges of order k = 2

and k = 3. Note that, since the number of possible hyperedges of order k is(Nk

), we

expect that the density of order k edges will decrease as k grows. Finally, to determine

the clustering in the hypergraph, we project the hypergraph onto a pairwise graph

such that the edge {i, j} exists if i and j are present in the same hyperedge. Then,

given this pairwise graph, we determine the community structure using the leading

eigenvector with the function cluster leading eigen() in the igraph package in R

(Csardi and Nepusz (2006)). We report the modularity of this clustering and the

number of clusters. The modularity measures the strength of the clustering and lies

within [−1, 1], where a high value indicates that the network can be divided clearly

into clusters. For each model, we have specified several cases which showcase the

features of the model and a summary of these cases can be found in Table 4.7.1.

Figures 4.7.2a, 4.7.2b and 4.7.2c show the results of study after 10,000 simulations

from each of the models. Clockwise from the top left, these plots show subgraph

counts for the motifs depicted in Figure 4.7.1a, percentiles of the hyperedge order,

the number of clusters and modularity, the density of hyperedges of order 2 and 3,

and percentiles of the node degree. We are able to demonstrate the strengths of each

model by comparing these metrics. For each model, the cases correspond to those

described in Table 4.7.1.

In Figure 4.7.2a we see the summary measures for the β hypergraph model. From

this bottom-left plot of this figure, we observe that each case demonstrates a very


different behaviour in the degree distribution as we would expect. In case 1, all nodes

have a similar degree and, in case 2, a portion of nodes have either a very large or

small degree. It is also apparent that, since case 2 generates denser hypergraphs,

larger motif counts are observed. The maximum hyperedge order was set to 4, and

so no hyperedges for k ≥ 5 are generated.

Figure 4.7.2b shows the equivalent plot for the model of Ng and Murphy (2018).

Firstly, we note that the topic clustering is consistently captured for all simulated

hypergraphs. We see that the degree distributions are largely the same, however by

comparing case 1 and 4 we observe that we can express different levels of connectivity.

When simulating from this model, we are unable to explicitly control the order of the

hyperedges and we note that this is controlled by the probabilities α and φ. We also

observe reasonably little variation in the motif counts and, for most cases, find that

triangles are more prevalent than the hypergraph motifs.

Finally, Figure 4.7.2c shows our results for the latent space hypergraph model.

Overall, we observe that there are a greater number of motifs observed than in the

previous models. Additionally, we see that we are able to demonstrate more control

over the motif counts. For example, consider case 3 where the order k = 2 hyperedges

are denser and in case 4 where the order k = 3 hyperedges are denser. The counts for

triangles and h1 subgraphs clearly reflect the number of hyperedges of each order. In

this model we also observe some control over the degree distribution and density. As

expected, when we increase the latent dimension to d = 3 and fix all other parameters

we obtain a sparser hypergraph and, to see this, compare cases 1 and 5. Finally, since

our graph is not designed to capture clustering, we do not observe consistent estimates

for the number of clusters. As commented previously, we may alter aspects of our

model to incorporate community structure or to vary the degree distribution. For

example, we may model the latent coordinates as a mixture of Gaussians.


Model Case Parameters

Stasi

et al.

(2014)

1) All nodes equally likely to

form connections

βi = −1.4 for i = 1, 2, . . . , N

2) Some nodes more likely to

form connections

β = (−0.5,−0.53, . . . ,−1.97,−2)

Ng and

Murphy

(2018)

1) Single hyperedge clusterG = K = 1, a = 1, φi1 = 0.075,π = b1, τ = 1

2) Distinct topic clusters only

G = 3, K = 1, a = 1, φi1 = 0.25 for

i ∈ A, φi2 = 0.25 for i ∈ B, φi3 = 0.25for i ∈ C, π = (1/3, 1/3, 1/3), τ = 1

3) Distinct size clusters onlyG = 1, K = 3, a = (0.2, 0.5, 1),φi1 = 0.15, π = 1, τ = (1/3, 1/3, 1/3)

4) Fuzzy topic clusters

G = 2, K = 3, a = (0.4, 1), φi1 = 0.3

for i ∈ A, φi2 = 0.3 for i ∈ B,φi1 = φi2 = 0.2 for i ∈ C,π = (1/2, 1/2), τ = (1/3, 1/3, 1/3)

LSH

1) Strongly correlated Σ

r = (0.18, 0.3, 0.35), µ = (0, 0),Σ = 0.25 ( 1 0.9

0.9 1 ),ψ0 = (0.01, 0.01, 0.01),ψ1 = (0.01, 0.01, 0.01)

2) No correlation in Σ

r = (0.18, 0.3, 0.35), µ = (0, 0),Σ = 0.25 ( 1 0

0 1 ) ,ψ0 = (0.01, 0.01, 0.01),ψ1 = (0.01, 0.01, 0.01)

3) Dense in e2, sparse in e3, e4

r = (0.2, 0.3, 0.35), µ = (0, 0),Σ = 0.25 ( 1 0

0 1 ) ,ψ0 = (0.01, 0.01, 0.01),ψ1 = (0.01, 0.5, 0.01)

4) Sparse e2, e4, dense in e3

r = (0.1, 0.35, 0.4), µ = (0, 0),Σ = 0.25 ( 1 0

0 1 ) ,ψ0 = (0.01, 0.01, 0.01),ψ1 = (0.01, 0.01, 0.01)

5) Increase latent dimension

from d = 2 to d = 3

r = (0.18, 0.3, 0.35), µ = (0, 0),

Σ = 0.25(

1 0 00 1 00 0 1

),

ψ0 = (0.01, 0.01, 0.01),ψ1 = (0.01, 0.01, 0.01)

Table 4.7.1: Cases for each hypergraph model considered in the model depth compar-

ison study. The case numbers correspond to the labels in Figures 4.7.2a, 4.7.2b and

4.7.2c. For all cases set N = 50 and, where appropriate, K = 4.


Currently, there do not exist theoretical results for subgraph counts within our

framework. The most relevant work can be found in Coulson et al. (2016), who

present results for subgraph counts in the SBM and graphons. However, it is not

obvious how to develop similar results for our framework.

In this study, we have clearly demonstrated the advantages of each modelling ap-

proach. It is clear that, for the construction we have chosen, our model presents a

flexible framework that is particularly appropriate for hypergraph data which exhibit

many motifs. However we may make alternative choices for the distribution of the

latent coordinates and the Cech complex to adapt our framework to express hyper-

graphs with different characteristics. We note that, whilst simulating these graphs

may be straightforward, fitting them may be much more challenging. This point will

be discussed further in Section 4.9.

4.7.2 Prior Predictive vs Posterior Predictive

In this section we examine the predictive degree distribution conditional on an ob-

served hypergraph. To explore the predictive distribution, we rely on the latent rep-

resentation to simulate new nodes and their associated connections given estimated

model parameters. Since the models of Stasi et al. (2014) and Ng and Murphy (2018)

contain node specific parameters, we comment that it is not immediately obvious

how to implement an analogue of this in either of their frameworks. We begin by

describing the study and set up, and then present our findings.

Using the latent space representation, we are able to examine how newly simulated

nodes connect to an observed hypergraph. Suppose that we have fitted the hypergraph

model detailed in Algorithm 8 to a hypergraph hobs and we obtain the parameter es-

timates µ, Σ, r, ϕ(0) and ϕ(1). Conditional on these estimates, we may simulate new

nodes and determine the hyperedges induced from these additional nodes. Through

repeated simulation we can then empirically estimate the predictive degree distribu-

tion of the newly simulated nodes and hobs.


● ●

●

●●● ●●●●●●●●● ●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●● ●●●● ●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●● ●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●● ●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●● ●●● ●●●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●

2.0

2.5

3.0

3.5

4.0

2.5%

25%

50%

75%

97.5

%

Per

cent

iles

Edge Size

● ● ●● ● ●● ●● ●●●●● ●● ●● ● ●● ●● ● ● ●●● ● ●● ●● ●●●● ●●● ●● ●●●●● ●●●● ● ●●● ● ●● ●● ●● ●●● ● ●●● ● ● ● ●●● ● ●● ●●● ● ● ●●●● ●● ●●● ● ●● ●● ●● ●

● ●●● ●●●●● ●● ●●●● ●●● ●● ●● ●●●● ●●●● ● ●● ● ●●● ● ● ●● ●●●●● ●● ● ● ●● ● ●●●

●●●● ●●● ●●●●● ●●● ●●● ●● ●●●●● ●●● ●●●● ● ●●●● ●● ● ●● ●●● ●● ●● ●●●● ●● ●●● ●●●● ●● ●● ● ●● ●●● ●● ● ●●●● ●● ●

●●●● ●● ●●●● ●●● ●● ●● ●●● ●● ●●●● ●● ● ● ● ●●● ● ●● ●● ●● ●● ● ●● ● ●● ●●●● ● ●● ●●● ● ●●●

● ●● ●●●● ●● ● ●● ● ●● ● ●●● ●●● ● ● ●

●● ●● ● ●● ●● ● ●●● ● ●● ● ● ● ●●● ●● ●● ●● ●● ● ● ●●● ● ●●● ●●● ●●● ● ●●● ●● ●●● ● ●●● ●●●● ● ●● ●●●

●● ●●●●● ●● ●● ●●● ●● ● ●●● ●●● ●●● ●● ●●● ●●● ●● ● ●●●● ●●● ● ●●●●

●● ●● ●●● ● ●●●● ●● ● ●●●● ●●● ●●● ● ●● ● ●●● ● ●●● ●● ●● ● ● ●●●● ● ●●●●●●●●● ●● ●●● ● ● ●● ● ●●

●● ●●●● ●●●● ●● ●● ●● ●● ● ●●● ●● ●●●● ● ● ●● ● ●● ● ●● ●● ●● ● ●● ●● ●●● ● ● ●● ●

● ●●● ● ● ●●● ● ●● ● ●● ●●● ●●● ●● ●● ●●● ●● ●● ●● ●● ●● ● ● ●●●● ●●● ● ● ●● ●● ●●●● ● ● ●●● ●● ● ● ●● ●● ●●● ●● ●

100

200

300

400

2.5%

25%

50%

75%

97.5

%

Per

cent

iles

Node Degree

●● ●● ●● ●● ●●● ● ●● ●● ●●● ● ● ●● ●●●● ●● ●●● ● ● ● ●●● ● ● ●● ● ●● ●● ●●● ●●●● ●● ●●● ● ●● ●● ●● ●● ● ●● ●● ●●●● ●● ● ●● ● ●●●● ● ●● ● ●●●● ●●●● ●●● ● ●● ●●● ●

● ●● ● ●● ●●● ●● ● ●● ● ●● ●●●● ●● ●● ● ●●● ●● ● ●●● ●●● ●● ●●● ●● ●● ●●● ●●●● ●● ● ● ●●●● ●●● ● ● ●● ●● ● ●●● ● ●● ●●●● ●●● ●● ●●●

●● ● ●● ● ●●● ● ●● ●● ●● ● ●● ● ●●●●● ● ● ●● ● ● ●●● ● ●● ●●● ● ●●●● ●● ●● ●●● ●● ●● ● ●●● ●● ● ●● ● ● ●● ●● ●● ●● ● ●● ● ● ●●●● ● ●●● ●● ●● ●● ● ● ●● ● ●● ●●● ● ●● ●● ● ●●● ● ●●● ● ●● ● ●●● ● ● ●● ●● ● ●● ●●● ●● ●● ●●● ●● ●●● ● ●●● ● ●●●●● ● ●● ●● ●● ●● ●● ●●● ● ●● ●●●● ●● ● ●●●● ● ●● ●● ●● ●●●● ● ●● ●●● ●●●● ● ●● ●●●● ●● ●●● ●●●● ●●●● ●● ● ● ●●● ●● ● ● ●

● ● ●●●● ●●● ●● ●●● ● ● ●● ●● ● ●●● ●●● ●●●● ●●● ●●● ● ● ●●● ●● ●●●● ●●●● ●● ●●● ● ●●●● ● ●●● ● ● ● ● ●● ● ●●●● ● ●●●● ●● ● ●●●● ●● ●● ●● ● ●●● ● ●● ● ● ●● ●●● ●●● ●●●● ●● ● ●● ●● ●●● ●● ●● ●● ●● ● ● ●●●● ● ●●● ●●●● ●●● ●●●● ● ●● ●● ●

050

00

1000

0 3 str

s4

strs

Sub

grap

hs

Count

●● ●● ●●● ●●● ●●● ● ●● ●●●●●●●● ●●●●●● ●● ●● ●● ●●● ●● ●● ●●● ●●●●●● ●●● ●●●●● ●● ●●●●●●● ●● ●●● ● ●● ●● ●●●●● ●● ●●● ●● ●●● ●● ●● ●● ●● ●●● ●● ●●● ●● ●● ●● ●●●●●●●●● ●●●●● ●●● ● ●● ● ●● ●●● ●● ●●●●●● ●● ●●● ● ●●● ●●● ● ●●● ●

● ●● ●●● ●●● ●●● ● ●● ●● ●● ●● ●● ●●●●●●● ●● ● ●●● ● ●● ● ●● ●●● ●●●● ●● ●● ● ●● ●● ● ●● ● ●●● ●● ● ●●●● ●●●● ●● ●● ●●●●● ●●●●●

● ●● ●●● ● ●● ●● ●● ● ●● ●●● ● ●● ● ●●● ●●●● ●●●● ● ●●● ●●● ●● ● ●●● ●● ●●

●●●● ●● ● ●● ●● ●● ● ● ●● ● ●● ●● ●●● ● ● ●● ●●● ●● ● ●●●● ●● ●●● ● ● ●● ●●● ●●● ● ●●● ● ●● ●● ●

●●●● ●●●● ●●●●● ●●● ●●●●●●●●●●●● ●●● ●●●● ●●● ●● ●●● ●●●● ●● ●●●●

●●●● ●● ● ●● ●● ●●● ● ●●● ●● ●● ●●●● ●●●●● ● ● ● ●● ● ● ●●●● ●●● ● ●●● ●● ● ●● ● ●● ●● ●●● ●● ● ● ●●● ● ●●●●●● ● ● ●●● ●●● ●●●● ●● ●● ●● ●● ● ●●●● ●

●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●● ●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●● ●●●●● ●●●●●●●●●●●●●●●●●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●● ●●●● ●● ●●●●●●●●●●● ●●●● ●● ●●●●●

050100

150

200

tris

h1

h2

h3

Sub

grap

hs

Count

−0.0

50

−0.0

250.

000

0.02

50.

050

mod

le

Mea

sure

Modularity

0.95

0

0.97

5

1.00

0

1.02

5

1.05

0 nc le

Mea

sure

Number of Clusters

case 1 2

●● ●● ●●●● ●● ● ● ●● ● ●●● ●●● ●●● ●● ●●● ● ● ●● ● ●●● ● ●● ●●● ●● ●● ● ●● ● ●● ● ●●● ● ●● ● ●● ●● ●● ●●● ● ●● ●● ●●●● ●

● ●●● ● ●●● ●●●● ●●● ●● ●● ●●● ●●● ●●● ●● ● ● ●●● ● ●

●● ●● ● ●● ●●● ●●● ●●●● ●● ●● ●●●● ●●● ●● ●●● ●● ● ●● ● ●● ●● ●●●● ●● ●●● ● ● ●● ●●●● ●● ●● ● ●●●●● ●●● ●● ● ●●●● ●●●● ●● ● ●● ●● ●●● ●● ●

●●● ● ●● ●● ● ● ●● ● ●● ● ●●●●●● ●● ●●● ●● ●● ●●● ●●● ●● ●● ●●● ●● ● ● ●● ●●●● ●● ●● ●● ●● ● ●● ● ●●● ●● ● ●● ●●●● ●● ● ●●

0.03

0.06

0.09

0.12

k = 2

k = 3

Hyp

ered

ge o

rder

Density

(a)β

-mod

elof

Sta

siet

al.

(201

4).

●

●●●●●

●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●● ●●●●● ●● ●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●

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5102.

5%

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Per

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iles

Edge Size

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200

300

400

2.5%

25%

50%

75%

97.5

%

Per

cent

iles

Node Degree

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0

5000

0

1000

00

1500

00

2000

003

strs 4

strs

Sub

grap

hs

Count

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020

040

060

0

tris

h1

h2

h3

Sub

grap

hs

Count

0.0

0.2

0.4

0.6

mod

le

Mea

sure

Modularity

1.0

1.5

2.0

2.5

3.0

nc le

Mea

sure

Number of Clusters

case 1 2 3 4

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● ● ●● ●● ● ●●● ● ● ●● ●●● ●●●●● ● ●● ●● ● ●●●● ● ●● ●● ● ● ●●● ●● ●●●●● ●●● ●● ●●●● ●● ● ●● ● ●

●● ●● ●●● ●● ●●● ● ● ● ●● ●● ●● ●● ● ●●● ●● ●●●● ●●●●● ●● ●● ●●● ●● ● ● ● ● ●●●●● ● ●●●● ●●● ● ●●● ●● ●●●● ●●● ● ●● ●● ●●●● ●● ●

● ●● ●●● ●● ●● ●●●● ●● ●● ●● ●● ●● ●●● ● ●●● ●●●● ●● ● ●● ● ●● ●●● ●●● ● ●●●●●●● ●●● ●● ● ● ●● ●●● ●● ● ●● ● ●●● ●● ●● ● ● ●●● ●

● ● ●●● ●● ●● ●● ●●●● ●● ● ●● ●●● ●● ●●● ●●● ● ●●● ●● ●● ●●● ●● ● ●● ●●●●● ●●● ● ● ●● ● ●●● ● ●●●● ●●● ●● ●●● ●● ●●● ●● ●●● ●● ●●●● ● ●● ● ●● ●●●●● ● ●● ● ●●●●●● ●● ● ●● ●● ●● ● ●● ● ●● ●●●● ●● ● ●●● ●●●● ●●● ●●

●● ●● ●● ● ●● ●●● ● ● ●● ●● ●●●●● ● ● ●● ●● ●●●●● ●●● ● ●●● ● ●● ●●●● ● ● ● ●● ●● ●●● ●●●● ●● ●●●●● ●● ●● ●● ●●●●●● ●● ●● ● ●●● ●● ● ●●●● ● ● ●●●● ●●●●●● ●● ●● ●● ●●●● ●● ●●● ● ●●● ●●● ● ● ●●●● ● ●●● ● ●●

0.0

0.1

0.2

0.3

0.4

k = 2

k = 3

Hyp

ered

ge o

rder

Density

(b)

Mod

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clu

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hyp

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mod

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Ng

and

Mu

rphy

(201

8).

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● ●●●●●●●●●● ●●●●●●●●●●●● ●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ●●●●●●● ●●●● ●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●● ●●●● ●●●●● ●●●● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●● ●●●●● ●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●● ●●●●●●● ●●●● ●●●●●● ●●●●●●●● ● ●●●●●●●●●● ●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●● ● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●● ●●●● ●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

2.0

2.5

3.0

3.5

4.0

2.5%

25%

50%

75%

97.5

%

Per

cent

iles

Edge Size

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025

0050

0075

002.

5%

25%

50%

75%

97.5

%

Per

cent

iles

Node Degree

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010

0020

00

tris

h1

h2

h3

Sub

grap

hs

Count

●● ●● ● ● ● ●● ●● ● ●● ●● ● ● ● ●●● ●● ● ●● ●●● ● ●● ●● ●● ● ●● ●●● ● ●● ● ● ●●● ●●● ● ● ● ●● ● ● ●● ● ●● ●● ●● ●● ● ●●● ● ● ●● ● ●●●

●● ●● ● ●●● ●● ●● ● ●● ● ●● ●● ●●●● ● ●●●● ●●● ●●●● ● ●●● ●● ●● ● ● ●● ● ●● ●● ●●● ●●● ●● ●●● ●● ● ●● ●●● ●● ●●● ● ●● ● ●● ● ●● ●● ●●● ● ● ● ●●● ●● ● ● ● ●●● ●●

●● ●● ●●●● ● ● ●● ● ●● ●● ● ●● ● ●●● ●● ●● ● ●●●●●● ●● ●● ● ●●●● ●● ● ●● ●● ●●● ●● ●●●● ● ● ●●● ●● ● ●● ●● ●● ●● ●● ●● ●● ●●●● ●● ● ●●●● ● ● ●● ●●● ●● ● ●●●

● ●● ●● ●● ●● ●● ●● ● ● ●● ● ●● ●● ●● ●● ●● ● ●●● ● ●● ●● ●●● ● ● ● ● ●● ●● ●●●●● ●●● ● ●●● ●● ●●●● ● ●● ●● ●●● ●●● ●● ● ●●● ●● ● ●● ● ● ●●●● ● ●● ●● ●

●● ●●● ● ●● ●● ●● ● ●● ●● ●●● ●● ●●● ● ●● ● ●●● ● ●● ●●● ●●● ●● ●● ●● ●● ●● ●● ● ●●●● ●● ● ● ●● ● ●●● ● ●● ●●● ●● ● ●● ● ● ● ●●

0.00

0.05

0.10

0.15

mod

le

Mea

sure

Modularity

●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●

●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●● ●●●●●●●●●●●

246

nc le

Mea

sure

Number of Clusters

case 1 2 3 4 5

● ●● ● ●●● ●●● ●● ●● ● ●●● ●●● ●● ●● ●●● ●●● ● ●● ●● ●●● ● ● ●● ● ●●●● ●● ●●● ● ● ●●● ●● ●● ● ● ●●● ●● ●●● ●●● ●● ●● ● ● ●●●● ● ●● ●●● ●●● ●● ●● ● ● ●●●●●● ●● ● ●●●●● ●● ● ●●●● ● ●●● ● ●● ●● ●● ●●●

● ●● ●●● ● ●●●●●● ●● ●● ●● ●●● ●●●● ●● ●● ● ●● ●● ● ●● ●● ● ●●● ●● ●● ● ● ●●●● ● ● ●●●● ●● ●● ●● ●●● ●●●● ●● ●● ● ●● ● ●● ●● ●●● ● ●●● ●● ● ●● ● ●●● ●●● ●●●● ●● ●●● ●●● ● ●●●● ● ●● ● ●●●●●● ●● ●● ●● ●●● ●● ●●●●● ●●● ● ● ●●●● ●●● ●● ●● ●●● ●● ●●● ●●● ● ●●●●●● ●●● ●●●● ● ●● ● ●● ●●● ●●● ●●● ●● ● ● ●●●● ● ●● ● ● ●● ● ●●●● ● ●● ● ●● ●● ● ●● ●● ● ●● ●● ● ● ●●● ●● ●●● ●● ● ● ●●●● ●● ●● ●●●●● ●● ●●● ●● ● ●● ●●●

●● ● ●● ● ● ●● ●●●● ●● ●● ●● ●● ●● ●● ●● ●● ● ●● ●●● ● ●● ● ●● ●●● ●● ●●●● ●● ● ●● ●● ●● ●●● ●● ●● ● ●●● ● ●● ●● ●● ● ●●●● ●●● ● ●● ●● ●● ●● ●●● ●●●● ● ●●●● ●●● ●● ●●● ●● ●●● ●● ●●● ● ●●●●●● ● ● ●● ●●●● ●● ●●● ●● ●●●● ●●● ● ●● ● ●● ●●● ●● ●● ●● ● ●● ●●● ● ●●● ●● ●● ● ● ●●● ●●●● ●● ●● ●●● ●● ● ●●● ● ●● ●● ●● ●● ●● ● ● ●●● ●●●● ● ●● ● ●●● ●●● ●●● ● ●●● ● ●●● ● ●● ● ●● ●● ●● ●● ●● ●●● ●●● ● ● ●● ● ● ● ● ● ●

●● ●● ●●● ● ●●● ●●● ● ●● ●●● ●● ●●●● ●● ● ●●● ● ●●● ● ●● ●● ●●● ●● ●●● ●● ●● ●● ● ●● ●● ● ●● ●● ●● ●● ● ●● ●●● ●● ●● ●● ●●● ●● ● ●● ●● ●●●● ●●● ●●● ●● ●● ●●● ● ●● ● ●● ●●●● ●● ●●●● ●●● ● ●●● ●● ● ●● ●●● ● ● ●●● ● ●● ●● ●● ●● ●●● ●● ●● ● ●●●● ● ●● ● ● ● ● ● ●● ●● ●●● ●● ●● ●● ●● ●● ●● ●● ●

● ●● ●●● ●● ●● ● ●● ● ●●● ● ●● ● ●● ●●● ●● ●● ● ●● ●●● ● ●● ●● ●● ●●● ●● ● ● ●●● ● ●● ●●● ● ● ● ●●● ●● ●● ●● ●● ● ●● ●●● ● ●●●● ●● ●● ●● ●●● ● ●● ●● ●● ●● ● ●●●● ● ●●● ●●● ●●● ●● ● ●●● ●● ●● ●●● ●● ●●● ●●● ●● ● ● ● ●●● ● ●● ● ●● ● ●●●● ● ●●● ● ●●●● ●● ●●● ● ●●● ●●●● ● ●● ●●● ●● ●● ● ● ●● ● ●● ●●● ●● ● ● ●● ●●● ●● ●● ● ●● ● ● ● ●●●

●● ●● ● ●●● ●● ● ●●●● ● ● ●●●● ●● ● ●● ●● ● ●● ●●● ●●● ●● ●● ●● ● ● ● ●● ●●● ●● ●●● ● ● ●●● ● ●● ●●●●●● ●●●● ●● ●● ●● ●●● ●● ● ●●● ● ●● ●●● ● ●● ●● ●● ● ●● ●● ● ●● ●● ●●● ● ●● ● ●● ●● ●●● ● ● ●● ● ●● ● ●●● ●● ● ●● ●● ●●● ● ●● ● ●● ●●● ●● ● ●● ●●● ●●● ●●● ●● ●● ●●● ●● ● ●● ● ● ● ●● ●●● ● ●● ● ●●● ●●● ●●● ●● ●● ●●● ● ●● ● ● ●● ●● ●●●●● ●

● ● ●●●● ●● ●● ●● ●●●● ●● ● ●●●● ●●●● ● ●●● ● ● ●●●●● ● ●● ● ●●●●● ●● ● ●●● ●● ● ●●● ●● ● ●● ●●● ●● ●● ●● ●●● ●● ● ●● ● ● ● ●● ●● ●● ● ●●● ● ●● ●● ●● ● ●●● ●●● ● ●● ●●● ●● ●●●● ● ● ●●● ●●●●● ● ● ●● ● ● ●● ●●●● ●●●● ● ●● ●●● ●● ● ●●● ●●● ● ●●● ● ●●● ●● ●● ●● ●●● ● ●●● ● ●● ● ●● ●●●● ● ●● ●● ● ●● ● ●● ● ●

●● ●● ● ●● ●● ● ●● ●● ●● ●● ●● ● ●●●● ●● ● ●● ● ● ●●● ● ●●● ●●● ●● ● ●● ● ●● ●● ● ●●● ●● ● ●● ● ●● ●● ●● ● ● ●● ●● ●●● ● ●● ● ●● ● ● ●● ● ●●● ●● ●● ● ●● ●●● ● ●●●● ● ●●●● ●● ● ●● ● ●● ● ●● ●● ● ●● ●● ●●● ●● ● ●● ●●● ● ● ●● ● ●● ●● ● ● ● ●● ● ●● ●● ● ● ● ●● ●●● ●●● ● ●●● ● ● ●●●●● ● ● ●● ●● ● ●● ●●●●● ●● ●● ● ●● ●●●● ●●● ● ● ●● ● ●●● ●● ● ●●● ●●● ●●●● ●● ●● ●● ●● ●● ●● ● ● ● ●●● ●● ●● ●● ●● ●● ● ●●● ● ● ●●● ●● ●● ●● ●● ●● ●●● ● ●●● ● ● ●●● ● ●● ● ● ●● ●● ●● ●●● ●●● ●● ●● ● ●●● ● ● ●● ●●● ● ●● ●●● ●● ● ●● ● ● ●

0.0

0.1

0.2

0.3

k = 2

k = 3

Hyp

ered

ge o

rder

Density

(c)

Lat

ent

spac

ehyp

ergr

aph

mod

eld

etai

led

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lgor

ith

m8.

Fig

ure

4.7.

2:Sum

mar

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hyp

ergr

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ula

ted

from

each

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em

odel

sco

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tion

4.7.

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4.7.

1.


To implement this procedure, we being by simulating a hypergraph hsim according

to Algorithm 8 with N = 50, d = 2, K = 3, r = (0.32, 0.4), ψ(0)k = ψ

(1)k = 0.001, µ =

(0.16, 1.24) and Σ =(0.58 00 0.58

). We then estimate the model parameters for this hy-

pergraph and, after 10,000 post burn-in iterations, we obtain r = (0.13, 0.16), ψ(0) =

(0.0058, 0.0014), ψ(1) = (0.0057, 0.0035), µ = (−0.13, 0.44) and Σ =(

0.14 −0.0039−0.0039 0.078

).

To estimate the posterior predictive degree distribution we apply the following

procedure nrep times.

1. Set u∗i = ui for i = 1, 2, . . . , N .

2. Simulate coordinates u∗i ∼ N (µ, Σ), for i = N + 1, N + 2, . . . , N +N∗.

3. Determine the hypergraph h∗sim obtained by taking the union of hobs and the

additional hyperedges induced from U ∗ = {u∗i }N+N∗

i=N+1, r, ψ(0) and ψ(1).

4. Calculate the degree distribution of h∗sim.

By averaging over the nrep = 100, 000 simulated degree distributions, we then ob-

tain an estimate of the degree distribution. Since we know the true model parameters,

we also estimate the true predictive degree distribution. We refer to this distribution

as the prior predictive below.

Recall that, for ui ∈ R2, two coordinates are specified as anchor points throughout

posterior sampling. The fixing of these points determines the scaling and hence affects

the magnitude of Σ and r. It is therefore not appropriate to compare parameter

estimates with the truth directly. However, by comparing the prior and posterior

predictive degree distributions, we consider a fair comparison between the true and

estimated model parameters. We expect that these two distributions are similar if

the model has been fitted well.

The full estimated prior and posterior predictive degree distributions are presented

in Figure 4.7.3, where the left panel shows the estimated prior predictive and the right

panel shows a qq-plot of the prior and posterior predictive degree distributions. We


0 2 4 6 8 11 14 17 20 23

Degree

Pro

babi

lity

0.00

0.02

0.04

0.06

0.08

●●●●●●●

●

●

●●

●●

●●●●

●● ●

●● ●

● ●

0.00 0.02 0.04 0.06 0.08

0.00

0.04

0.08

'Truth'

Est

imat

e

(a) Predictive degree distributions for hyperedges of order k = 2.

0 23 49 75 105 139 173 207 241

Degree

Pro

babi

lity

0.00

00.

010

0.02

00.

030

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●

●

●●

●

●

●

●

0.000 0.010 0.020 0.030

0.00

00.

010

0.02

00.

030

'Truth'

Est

imat

e

(b) Predictive degree distributions for hyperedges of order k = 3.

Figure 4.7.3: Comparison of prior and posterior predictive degree distributions for

N∗ = 10 newly simulated nodes. In each figure, the left panel shows the prior predic-

tive degree distribution, and the right panel shows a qq-plot of the prior and posterior

predictive degree distributions. Figures 4.7.3a and 4.7.3b show the degree distribu-

tions for hyperedges of order 2 and 3, respectively.

see that there is a strong correspondence between the two distributions, particularly

for the order k = 2 hyperedges. However, there is a slight difference in the upper tail

for hyperedges of order k = 3. This may be due to the complexity of the space or the

greater number of constraints placed on the latent coordinates of highest degree.

We may also examine the distributions for hyperedges occurring between newly

simulated nodes only, and the hyperedges occurring between the nodes of hsim and the


Index U misspecification r misspecification

1 None None

2 ui ∼∑C

c=1 λcN (µc,Σc), C = 2 and distinct

clusters

None

3 ui ∼∑C

c=1 λcN (µc,Σc), C = 2 and fuzzy

clusters

None

4 ui sampled from a homogeneous Poisson

point process

None

5 None Simplicial: rk = rk−16 ui ∼

∑Cc=1 λcN (µc,Σc), C = 2 and distinct

clusters

Simplicial: rk = rk−1

7 ui ∼∑C

c=1 λcN (µc,Σc), C = 2 and fuzzy

clusters


8 ui sampled from a homogeneous Poisson

point process


Table 4.7.2: Types of misspecification.

newly simulated nodes. Plots for these cases are presented in Appendix B.7 and, for

both cases, we see a close correspondence between the prior and posterior predictive

degree distributions.

4.7.3 Misspecification

In this section we consider the robustness of our modelling approach under different

cases of misspecification. This allows us to assess the suitability of our model for a

range of hypergraph data which do not satisfy our modelling assumptions. We begin

by detailing the types of misspecification we consider and then we describe the set up

for the simulation study. Finally, we present our results.

In the specification of our model, we assume behaviour on the latent coordinates,

the radii and the noise parameters which control the hyperedge modifications. To

design the study, we consider alternative mechanisms which may have generated the

observed hypergraphs. The types of misspecification we consider for each of these


parameters are detailed in Table 4.7.2. We comment here that alternative cases

of misspecification would be interesting to explore, such as node specific radii or

non-homogeneous errors, however there are practical limitations which need to be

addressed.

For each of the 8 cases described in Table 4.7.2, we simulate a hypergraph with

N = 50 and K = 3 and fit our model. Then, using a similar procedure to the one used

in Section 4.7.2 with N∗ = 10, we explore the predictive distributions. We compare

the predictives obtained by simulating from the true model and the estimated model.

In particular, we record

• The degree distribution.

• The number of occurrences of the motifs depicted in Figures 4.7.1b3, 4.7.1a3,

4.7.1a4 and 4.7.1a5.

• The density of order 3 hyperedges.

The results of our study are presented in Figure 4.7.4, where the ith row corre-

sponds to the ith case detailed in Table 4.7.2. We generally observe a close corre-

spondence between the prior and predictive distributions, however we see an overall

poorer performance when the simulated hypergraphs are simplicial. It is interesting

to note that, although there is generally a good fit in terms of the degree distribution,

we see many of the motifs are over or under predicted by the posterior predictives.

This reflects the constraints implied by the latent representation. From this study

it is clear that our model imposes certain properties on the predictive distributions

and it is important to verify whether or not these are appropriate for a specific data

example.


Figure 4.7.4: Summary of misspecification simulation study. Left to right: average

degree distribution, number of triangles Figure 4.7.1b3, number of Figure 4.7.1a3,

number of Figure 4.7.1a4, number of Figure 4.7.1a5 and density of order 3 hyperedges.

Each row corresponds to the misspecification cases summarised in Table 4.7.2. The

y and x axes show the quantiles of the posterior and prior predictives, respectively.

The red lines correspond to y = x.


4.8 Real data examples

In this section we analyse three real world hypergraph datasets using the model we

have introduced. To begin, we examine a dataset constructed from actor co-occurrence

in ‘Star Wars: A New Hope’ and compare our analysis with that of Ng and Murphy

(2018). Next, we consider a dataset describing company leadership and investigate

predictive inference on the observed hypergraph compared to hypergraphs imputed

from the pairwise projected graph. Finally, we consider a coauthorship dataset and

asses predictive inference by comparing to the observed hypergraph.

4.8.1 Star Wars: A New Hope

In this section we consider a dataset constructed from the script of ‘Star Wars: A New

Hope’ which describes co-occurrence between the eight main characters. We represent

this as a hypergraph where the nodes represent characters and hyperedges indicate

which characters appeared in a scene together. In this dataset we have N = 8 and

K = 4, and we remove repeated hyperedges and hyperedges of order one to ensure

the data is amenable to analysis under our model. This dataset was considered in Ng

and Murphy (2018), and we compare and contrast our methodology to this approach.

Recall that, in our model, observed hyperedges can be explained by the latent

geometry or the hyperedge modification. To ensure most hyperedges are explained

by the latent representation, we fix an upper limit for the parameters ϕ. In doing so,

we encourage interpretable latent coordinates and improve the quality of predictive

inference. To begin, we fit the model detailed in Algorithm 7 and set the upper limit

for ϕk to be 0.75× the density of order k hyperedges, for k = 3, 4.

The posterior mean of the latent coordinates after 37500 post burn-in iterations

is given in Figure 4.8.1a. In this figure, orange lines indicate a pairwise connection,

and blue and purple regions correspond to order 3 and 4 hyperedges, respectively.

We observe a group of well-connected nodes which contains the character “Luke” in


●

●

●

●

●

●

●

●

−1.0 −0.5 0.0 0.5

0.0

0.1

0.2

0.3

uil

u i2

●

●

●

●

●

●

●

●

WEDGEC−3PO

LEIA

BIGGS

HAN

LUKE

DARTH VADER

OBI−WAN

(a) Upper limit of ϕk is set to 0.75× the

observed order k hyperedge density.

●

●

●●

●

●

●

●

−1.0 −0.5 0.0 0.5 1.0

−1.

5−

1.0

−0.

50.

00.

51.

01.

5

uil

u i2

●

●

●●

●

●

●●

HAN

LUKE

DARTH VADER

LEIA

OBI−WAN

C−3PO

WEDGE

BIGGS

(b) Upper limit of ϕk is set to 1.5× the

observed order k hyperedge density.

Figure 4.8.1: Posterior mean of latent coordinates for the Star Wars dataset for

different upper limits on ϕ. Connections in orange, blue and purple correspond to

hyperedges of order k = 2, 3 and 4, respectively.

its centre. In Ng and Murphy (2018), this character was highlighted to be likely to

occur in the two largest topic clusters, and we comment that the latent representation

reflects this characters importance. Note that a similar observation is made in the

network visualisation literature, where nodes with a greater number of connections are

placed more centrally. The main group of nodes in Figure 4.8.1a is largely determined

by the order 3 and 4 hyperedges between “Leia”, “C-3PO”, “Luke”, “Obi-Wan” and

“Han”. We note that the characters “Wedge” and “Darth Vader” are less connected,

and so we see them located on the periphery of the latent representation.

We also consider setting the upper limit for ϕk to be 1.5 × the density of order

k hyperedges, for k = 3, 4, and the posterior mean of the latent coordinates for this

case is given in Figure 4.8.1b. Here we also see the importance of the character

“Luke” reflected in the latent coordinates, however the increased noise parameter


0.00

0.25

0.50

0.75

1.00

2 4 6 8

Degree of Luke

Pro

port

ion

0.00

0.25

0.50

0.75

1.00

0 2 4 6

Degree of Darth Vader

Pro

port

ion k

2

3

4

(a) Upper limit of ϕk is set to 0.75× the observed order k hyperedge density.

0.00

0.25

0.50

0.75

0 5 10 15 20

Degree of Luke

Pro

port

ion

0.00

0.25

0.50

0.75

1.00

0.0 2.5 5.0 7.5

Degree of Darth Vader

Pro

port

ion k

2

3

4

(b) Upper limit of ϕk is set to 1.5× the observed order k hyperedge density.

Figure 4.8.2: Predicted degree distributions conditional on the fitted model. Given

U , r and ϕ we simulate the connections in the hypergraph to estimate the degree

distribution, and the upper limit for ϕ is 0.75 the hyperedge density in Figure 4.8.2a

and 1.5 the hyperedge density in Figure 4.8.2b. The left plots show the degree distri-

bution for “Luke” and the right plots show the degree distribution for “Darth Vader”.

The order 2, 3, and 4 hyperedges are shown in green, orange, and purple, respectively.

means that the latent coordinates are less constrained. To make this point further,

we now consider the variability in the observed connections.

As mentioned in Section 4.7, our modelling framework allows us to explore pre-

dictive distributions. Given the fitted model, we can simulate new connections to

examine how variable the degree of specific nodes are expected to be. More specif-

ically, we consider the degree distributions for the characters “Luke” and “Darth

Vader” by repeatedly simulating their connections given U , r and ϕ. We do this for


both choices of upper limits on ϕ.

The results of this for upper limits of 0.75 and 1.5 times the hyperedge densities

are shown in Figures 4.8.2a and 4.8.2b, respectively. In both Figures, we see a clear

difference between the levels of connectivity for each of these characters. Since “Luke”

is more centrally located with respect to the latent coordinates, we observe that this

characters is expected to be more connected than “Darth Vader” who is located on

the periphery. This tells us that nodes which are more centrally located are expected

to be more connected in the hypergraph. Comparing Figures 4.8.2a and 4.8.2b, we

more variability and a higher level of connectivity in Figure 4.8.2b. This is due to

the larger upper limit on ϕ. Since the noise is estimated to be larger, the latent

representation explains fewer of the observed hyperedges. Therefore, when we fix the

coordinates to U , our estimates may not reflect the observed hypergraph well.

Although we are able to draw parallels between the our observations and the

observations of Ng and Murphy (2018), our approach differs considerably. We now

comment on the advantages of each approach. Firstly, in Ng and Murphy (2018) the

authors incorporate multiple occurrences of a hyperedge and hyperedges containing

a single node into their analysis. Our methodology does not facilitate this, and so

the dataset was reduced accordingly. Secondly, our model provides a visualisation

of the hypergraph and the approach of Ng and Murphy (2018) does not. When the

parameters ϕ are appropriately constrained, this visualisation reflects many observa-

tions of the analysis in Ng and Murphy (2018). Finally, our framework allows us to

investigate the predictive distributions, and this has allowed us to comment on the

expected variability of the degree of certain nodes.

4.8.2 Corporate Leadership

In this section we consider a dataset describing person-company leadership in which

individuals are represented as nodes and hyperedges connect individuals who have

held a position of leadership in the same company. This dataset is available from


KONECT (2017), and to analyse this dataset we remove hyperedges of order larger

than 4. We comment that our model can in theory express hyperedges of arbitrary

order, however larger values of K become increasingly computationally challenging in

our framework. Finally, we remove nodes with degree 0 to obtain a hypergraph with

N = 17 nodes.

As noted in Section 4.1, many hypergraph datasets are analysed in terms of the

graph obtained by connecting nodes i and j if they are contained within the same

hyperedge. This results in a loss of information and in this section we consider the

effect of this in more detail. There are a multitude of ways in which we can assume

the hypergraph connections given the graph, and we focus on two in this section. We

consider the hypergraph obtained by representing each maximal clique as a hyperedge

and the simplicial hypergraph obtained by representing each clique by a hyperedge.

For convenience, we refer to each of these hypergraphs as the maximal clique hyper-

graph and simplicial clique hypergraph, respectively. To investigate the effect of these

choices on predictive inference, we fit the model detailed in Algorithm 7 to the hyper-

graph with K = 4, the projected graph with K = 2, the maximal clique hypergraph

with K = 4 and the simplicial clique hypergraph with K = 4. As in Section 4.8.1, we

constrain ϕk to be at most 20% of the observed density of order k hyperedges.

The posterior mean of the latent coordinates for each hypergraph are shown in

Figure 4.8.3. We observe several similarities between the latent representations, such

as the placement of the set of nodes {3, 4, 15, 12}, however the constraints imposed

on the latent coordinates differ for each case. For instance, comparing Figures 4.8.3a

and 4.8.3b, we see that the hyperedge {1, 8, 14} is expressed differently due to the

respective geometric constraints. Furthermore, there is a clear difference between the

density of order 2 hyperedges in these cases which will have a significant impact on

predictive inference. For this dataset, the maximal clique hypergraph is very similar

to the observed hypergraph and we note that this will not be true in general. In

fact there exist many possible hypergraph whose projected graph corresponds to the


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Figure 4.8.3: Posterior mean of the latent coordinates after 25000 post burn-in itera-

tions. Figure 4.8.3a: observed hypergraph with K = 4. Figure 4.8.3b: graph obtained

by connecting nodes if they are contained within the same observed hyperedge. Fig-

ure 4.8.3c: hypergraph obtained from representing maximal cliques in the graph by

a hyperedge. Figure 4.8.3d: simplicial hypergraph obtained by representing cliques

in the graph by a hyperedge. Connections in orange, blue and purple correspond to

hyperedges of order k = 2, 3 and 4, respectively.

graph in Figure 4.8.3b. Finally, we comment that the simplicial clique hypergraph in

Figure 4.8.3d imposes the strongest constraints on the latent representation due to

the density of order 2 and order 3 hyperedges.


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Figure 4.8.4: Predictive distributions for motif counts for N∗ = 5 newly simulated

nodes. Top row: predictives for the motifs shown in Figures 4.7.1a3, 4.7.1a4 and

4.7.1a5. Bottom row: predictives for the motifs shown in Figures 4.7.1b1, 4.7.1b2 and

4.7.1b3.

To further compare these fitted models we consider predictive inference for N∗ = 5

newly simulated nodes. We focus on the motifs shown in Figure 4.7.1b and Figures

4.7.1a3, 4.7.1a4 and 4.7.1a5, and the predictive motif counts are shown in Figure 4.8.4

for each model fit. First, we will discuss in predictive inference for the motifs involving

order 2 edges, namely m1 (Figure 4.7.1b1), m2 (Figure 4.7.1b2) and triangles (Figure

4.7.1b3). We see a clear similarity between the predictives for the graph and simpli-

cial clique hypergraph, and the observed hypergraph and maximal clique hypergraph.

As commented above, this can be explained by the particular characteristics of the

observed hypergraph and this will not be the case in general. It is important to note

that the edges in the graph and the order 2 hyperedges in the hypergraphs have a dif-

ferent interpretation, and so it may not be appropriate to make a direct comparison.

However, it is clear that there is a large difference between the number of predicted

motifs in for the observed hypergraph and the graph. We now consider the motifs in-

volving order 3 edges, namely h1 (Figure 4.7.1a3), h2 (Figure 4.7.1a4) and h3 (Figure

4.7.1a5). Since the graph contains no information about hyperedges, we cannot pre-


dict the occurrence of these motifs. There is a large difference between the predictives

for the maximal clique hypergraph and the simplicial clique hypergraph. Generally

speaking, it is not possible to make accurate inference about the observed hypergraph

from these hypergraphs. However, we see a similarity between the maximal clique hy-

pergraph and observed hypergraph predictives in this case. Since our model expresses

non-simplicial hypergraphs, we have non zero predictives for the motifs h1 and h2 for

each hypergraph dataset. However, in the simplicial clique hypergraph, we find that

the motif h1 is relatively less likely to occur. This reflects a tendency for simplicial

relationships.

4.8.3 Coauthorship for Statisticians

In this section we return to our motivating example of coauthorship. We consider the

dataset provided by Ji and Jin (2016) which describes coauthorship between N = 3606

statisticians where K = 10. Analysing the full dataset within our framework is

computationally prohibitive, and so we consider a subset with N = 48 and K = 3. To

obtain the subset we first restrict to hyperedges of order less than 4. Then we select

a seed node and include hyperedges involving this node with probability p = 0.9.

We repeat this process with the nodes added to the hypergraph and we refer to each

newly added set of nodes as a ‘wave’. To maintain a reasonably sized hypergraph

subsample, we decrease the probability of inclusion by 0.15 for each new ‘wave’. Our

interest is in assessing the quality of predictive inference by comparing the predictives

to the next sampling ‘wave’.

We fit the model detailed in Algorithm 7 and the posterior mean of the latent

coordinates is shown in Figure 4.8.5a. As in the previous examples, we restrict the

hyperedge modification to be at most 20% of the observed hyperedge density. By

comparing the hyperedges in the graph gN,K(U , r) to those present in the observed

hypergraph, we find that 96% of order 2 hyperedges and 99% of order 3 hyperedges are

explained by the latent representation. Hence, the hyperedge modification explains


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(a) Posterior mean of latent coordinates

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peredges of order 2 and 3 are shown in

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(b) Observed degree distribution com-

pared to theoretical degree distribution

calculated using the fitted model param-

eters. We obtain pe3 through simulation.

Figure 4.8.5: Fitted model for coauthorship example.

only a small proportion of the observed hyperedges. We note that the nodes 1, 7 and

25 have the highest degree, and we see these nodes placed close to the estimated mean

µ of the distribution of the latent coordinates. For this example, we also consider the

theoretical degree distributions discussed in Section 4.6. In Figure 4.8.5b we compare

the observed degree distribution with the theoretical degree distribution calculated

with the fitted model parameters. Since we cannot find an exact expression for pe3 ,

we obtain an estimate of this using simulation. We observe a strong correspondence

between the observed degree distribution and the degree distribution predicted by the

theory.

To further asses the fitted model, we now compare subgraph counts for the next

wave of sampling with the predictive distributions. We estimate the predictive distri-

bution for the subgraphs shown in Figures 4.7.1a3, 4.7.1a4, 4.7.1a5, 4.7.1b1, 4.7.1b2

and 4.7.1b3, and compare this with the counts observed in the next wave of subsam-

pling. The predictive distributions for the addition N∗ = 19 nodes are shown in Figure


4.9.1, where the subgraph counts observed in the next wave of sampling are shown

in red. Due to the subsampling mechanism, we expect the next 19 nodes to mostly

be placed on the periphery of the point cloud. To mimic this, we sample N + N∗

nodes from N (µ, Σ) and take the N∗ coordinates which lie furthest from µ in terms of

Euclidean distance. In Figure 4.9.1, we see a reasonable correspondence between the

observed and predicted, however we note that many of the observed subgraph counts

lie in the lower tail of the predictive.

4.9 Discussion

In this chapter we have introduced a latent space model for hypergraph data in which

the nodes are represented by coordinates in Rd. To extend the framework introduced

in Hoff et al. (2002), we have relied on a modification of a nerve construction which

allows us to express non-simplicial hypergraphs. This application of a nerve draws a

connection between stochastic geometry and latent space network models, and allows

us to develop a parsimonious hypergraph model. The latent representation imposes

properties on the hypergraphs generated from our model, including a type of ‘higher-

order transitivity’. This property, in which a presence of an order k hyperedge is

more likely given the presence of subsets of the hyperedge, is highlighted in the model

depth simulations in Section 4.7.1. In particular, we see a greater presence of certain

subgraph counts in comparison to the two other models considered. It is important

to note that, depending on the modelling choices, particular hypergraph relationships

may be challenging to represent using our nerve construction. For example, the maxi-

mum number of possible leaves in a star will be limited by the dimension of the latent

space. However this may be mediated by either choosing a different convex set to

generate the nerve, increasing the probability of hyperedge modification or adopting

a different specification for the latent positions.

The modification of the indicators for the hyperedges has two main motivations.


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Figure 4.9.1: N∗ = 19 predictive subgraph counts. Left to right: motifs depicted in

Figure 4.7.1a3, 4.7.1a4, 4.7.1a5, 4.7.1b1, 4.7.1b2 and 4.7.1b3. The red dots correspond

to the observed motif counts for the newly sampled nodes.

Firstly, without this modification, the conditional distribution L(U , r;hN,K) would

be equal to one only when there is a perfect correspondence between the observed and

estimated hyperedges. Hence model fitting may be difficult. Secondly, the modifica-

tion extends the support of the model to the space of all hypergraphs and, without

the modification, it is unclear whether an observed hypergraph is expressible within

our framework which greatly limits the applicability of our model. We note here that

techniques such as tempering can be used to aid model fitting, but the challenge of

characterising the support of the model still remains. From (4.4.1) we observed that

a hypergraph generated from our model is a modification of a nsRGH and, as the

probability of modification grows, the generated hypergraphs will behave more like

the hypergraph analogue to an Erdos-Renyi random graph. Therefore, to maintain

the hyperedge properties inherited from the latent space, we impose that the proba-

bility of modification is small. For a hypergraph on N nodes, the density of order k

hyperedges is given by the fraction of possible order k hyperedges that are present in

the hypergraph. It is clear that, to obtain a similar density of hyperedges for different

k, the probability of modification must scale according to number of possible hyper-

edges of each order. This means that the magnitude of the modification probability


will decrease as k grows. The hyperedge modification has a connection with mea-

surement error models in which the observations are assumed to be measured with

some noise. In the context of networks, Le et al. (2018) investigate the recovery of

an underlying true network given a set of noisy observations. Whilst this differs from

our setting, we can view our model in a similar way in which the nsRGH represents

the truth. This helps motivate our observation on the magnitude of the probability

of hyperedge modification.

To obtain posterior samples we rely on a Metropolis-Hastings-within-Gibbs MCMC

scheme in which each parameter is sampled conditionally on the remaining parameters.

In Section 4.4, we observed that the conditional distribution p(gN,K(U , r)|µ,Σ, r)

is invariant to rotations, translations and reflections of the latent coordinates U .

However, since samples are obtained from the conditionals, these sources of non-

identifiability can be removed in a post-processing step using a Procrustes transform.

This approach is typically used for latent space network models and ensures that the

samples have a clear interpretation. We instead infer the latent representation on the

Bookstein space of coordinates, which avoids the need for post-processing and further

removes the source of non-identifiability from joint rescaling of U and r. To initialise

the MCMC, we rely on techniques commonly used in the latent space network lit-

erature. Since random initialisation of U and r performs poorly, we use generalised

multidimensional scaling (GMDS) to determine initial values of U and the radii are

then scaled accordingly. GMDS was used in this context by Sarkar and Moore (2006),

and details of the MCMC initialisation are given in Appendix B.4. We also exploit

the connection with computational topology in our MCMC scheme. By relying on

existing tools we are able to sample from our model and, in particular, to calculate

the Cech complex we use the GUDHI C++ library (see The GUDHI Project (2015)).

Although the model we present can in theory be extended to hypergraphs with

arbitrary maximum edge size K, this proves computationally challenging in practice.

In our examples, we have predominantly restricted to cases in which K is equal to 3


or 4. This allows insight on the hypergraph relationships, but this restriction of K is

currently a large limitation of our methodology. In the next paragraph we comment

on a possible direction for a more scalable model. As previously mentioned, there are

some motifs, for example stars, which may be difficult to express in our framework.

Whilst the hyperedge modification can aid fitting here, increasing the probability of

modification may have undesirable effects on the other connections. The number of

leaves that can be expressed in a star is also related with the choice of latent dimension

(see Helly’s Theorem, Section 3.2 of Edelsbrunner and Harer (2010)). For interpre-

tation, we have assumed that d is small and choosing d in a more principled manner

requires careful consideration. In this chapter we have presented two different models

(detailed in Algorithms 7 and 8) for hypergraph. The model given in Algorithm 7

has a single modification parameter for each order hyperedge, and this model has a

clear interpretation. From Proposition 3.1 in Lunagomez et al. (2019), it follows that

hypergraphs with a greater number of modifications are less likely to occur. How-

ever, the interpretation for the model in Algorithm 8 is less straightforward. Since

there are competing sources of modification, the above argument only applies when

the density of the hypergraphs remain fixed. A related framework is considered in

Le et al. (2018), where the authors consider recovery of a true network given a set

of noisy realisations. In this work, the sources of edge noise are divided into false

positives and false negatives. By viewing the underlying nsRGH gN,K(U , r) as the

truth, we see that the probabilities of hyperedge modification in Algorithm 8 have an

analogous interpretation.

There are a number of extensions to the model we have presented that can be

explored. For instance, the choice of underlying distribution on U will affect the

characteristics of the hypergraphs expressed by the model. This can be examined

from a theoretical perspective by adapting the arguments discussed in Section 4.6,

or practically by simulating from the generative model. Exploring this would further

explain which aspects of the model depth simulations in Section 4.7.1 are an artefact


of our modelling choices. In Spencer and Rohilla Shalizi (2017) the authors assume

that the latent coordinates in a latent position are generated according to a Poisson

process, and show that this is able express graphs with various levels of sparsity. It

would be interesting to examine this within the hypergraph setting, and we leave

this to future work. Another extension would be to consider modelling hypergraphs

which exhibit community structure. One approach would be to assume that the latent

coordinates are distributed according to a mixture of Gaussians, an idea which has

been explored in the latent space network literature (for example, see Handcock et al.

(2007)). Alternatively, Rubin-Delanchy et al. (2017) show that a generalisation of the

latent position framework in which connections are determined via a dot product is

able to express graphs with community structure. This idea may also be considered in

the hypergraph setting, though this extension is likely to be more involved. There also

exist several other interesting extensions for which the adaptation of our model is less

clear. In many real world examples, the hyperedges may occur multiple times. This

motivates developing a model for non-binary hypergraphs, though it is not clear how

to extend our model accordingly. Another line of future work stems from considering

alternatives to the Cech complex. One choice to consider is the Vietoris-Rips complex

(Vietoris (1927), Gromov (1987)) in which an order k hyperedge exists if each of the(k2

)balls of radius r intersect. Evaluating this complex only requires considering the

pairwise intersections, and so is likely more scalable in terms of K. Finally, in this

chapter we have not explored the addition of covariate information. Typically covari-

ate information is included through an autoregressive term and a similar approach

may be explored here.

Chapter 5

Conclusions and Further Work

The contributions of this thesis focus on computational and modelling aspects of

the latent space approach for network data. Chapter 3 considered estimation of

temporally evolving latent space networks via SMC and Chapter 4 introduced a latent

space model for hypergraph data in which interactions occur between sets of nodes.

The methodology explored in this thesis can be extended in a variety of ways, some

of which were outlined in the conclusions of Chapters 3 and 4, and in this chapter we

highlight three possible directions for future work.

Improving computational scalability

A key computational challenge associated with the DLSN model considered in Chap-

ter 3 is the evaluation of the O(N2) terms in the likelihood. Estimating the latent

representation sequentially via SMC improves the scalability in terms of the number

of observations in time T , however this reduces the scalability in terms of the number

of nodes N . This is primarily due to characteristics of SMC methodology, but this

issue is further compounded by the computational cost of evaluating the likelihood.

To improve the scalability in terms of N we explored the high-dimensional SMC lit-

erature, and further improvements to the scalability may also be made by considering

likelihood approximations to reduce the O(N2) cost. In the existing literature, it is

116

CHAPTER 5. CONCLUSIONS AND FURTHER WORK 117

typical to rely on likelihood approximations (see Raftery et al. (2012) and Rastelli

et al. (2018)) to facilitate estimation of networks with larger N within an MCMC

scheme. Usually, each latent coordinate is sampled conditional on all other latent

coordinates via a MH-within-Gibbs update and, since the entire latent representation

is estimated jointly within an SMC scheme, a direct application of existing likelihood

approximations may not be appropriate. It is also important to highlight that, due to

mechanics the of SMC, employing a likelihood approximation may present additional

challenges that are not found within the MCMC context. Approximations within

SMC have been considered for static models within Gunawan et al. (2018) and the

adaption to the time varying case may not be straightforward.

Alternatively, we may explore a model-based approach to improving the scalability.

For example, Fosdick et al. (2019) introduce the latent space SBM in which the

within and between community connection probabilities are modelled via a latent

space model and an Erdos-Renyi random graph, respectively. By partitioning the

latent coordinates into distinct communities, the likelihood can be decomposed and

the distance calculation between all(N2

)pairs can be avoided. Additionally, it may be

possible to take advantage of the model structure to design a more efficient particle

filter which accounts for the regions of independence.

In the hypergraph model introduced in Chapter 4, we avoided the O(Nk) likelihood

terms implied by a construction analogous to that of Hoff et al. (2002). This was

achieved by viewing the hypergraph as an Erdos-Renyi modification of a construction

based on the Cech complex. This reduced the computational cost associated with

the likelihood calculation, however evaluating the likelihood relies on calculating the

Cech complex. The Vietoris-Rips (VR) complex (Vietoris (1927), Gromov (1987))

likely presents a more scalable alternative to the Cech complex, and this would be

an interesting extension to explore. In the VR complex an order k hyperedge is

present when each of the(k2

)balls of radius r intersect, meaning that no additional

computational cost is required to determine all hyperedges once the order 2 hyperedges


are known. However, altering the underlying complex affects the properties of the

resulting hypergraphs and an analysis of this should also be considered.

The hypergraph model presented in Chapter 4 can be adapted in many ways, sim-

ilar to the extensions of the initial models of Hoff et al. (2002) discussed in Chapter

1. However, it is important that certain properties of the model are preserved when

developing these extensions. For example, it is natural to include covariate informa-

tion into the model so that nodes which share similar covariates are more likely to

be connected. In the construction of Hoff et al. (2002), this can be achieved by in-

cluding covariate information in the linear predictor. The analogous approach in the

hypergraph setting would require a calculation over all order k sets, and this is com-

putationally expensive. Therefore, this must be included in a way which maintains

the computationally appealing properties of the likelihood.

Changepoint and anomaly detection

When a series of interactions are observed through time, it is natural to ask whether or

not the patterns of connectivity change. In the statistical literature this is referred to

as changepoint detection, where the objective is to determine the points at which the

characteristics of a time series change. This literature is well developed for general

time series data (for example see Truong et al. (2018), Aminikhanghahi and Cook

(2017) and references therein) and particular examples of changepoint detection for

temporally evolving networks include Wang et al. (2017), De Ridder et al. (2016) and

Ludkin et al. (2018). In the context of network data, changes may, for instance, relate

to the density of the network, the tendency for specific nodes to form connections

or the community memberships of each node. It would be interesting to consider

this problem in the context of latent space network models. Park and Sohn (2018)

develop a changepoint approach for networks using the tensor regression framework

of Hoff (2011), and this is the most relevant existing methodology. We also note here

that SMC presents a natural setting for online changepoint detection which has been


discussed in Heard and Turcotte (2017) which also presents an interesting direction

for future work.

The related problem of anomaly detection (see Patcha and Park (2007)), in which

the goal is to determine which observations are uncharacteristic of the overall time

series, may also be explored. Recently, this has been considered in the latent space

framework by Lee et al. (2019), where the authors develop a sequential variational

Bayes approach to inference. Variational methods consider a computationally cheaper

approximation to the target, whereas SMC methods do not. Hence, SMC presents an

interesting alternative which facilitates online inference.

Here we have focused on the DLSN model from Chapter 3, but it is worth noting

that the hypergraph model from Chapter 4 may also be extended to the dynamic

setting and considered within the problem of changepoint and anomaly detection.

However, it is important to note that this extension may not be entirely without

practical challenges.

Modifying the underlying geometry

The models considered in this thesis assume that the nodes can be appropriately rep-

resented in Euclidean space. This imposes properties on the resulting networks and,

depending on the application, this may be a major limitation of this work. Several

authors have explored the effect of modifying the underlying geometry within latent

space network models (see Krioukov et al. (2010), Smith et al. (2017) and McCormick

and Zheng (2015)). A particularly appealing choice is hyperbolic geometry since Kri-

oukov et al. (2010) demonstrate that this naturally represents networks with power

law degree distributions. Adapting the DLSN from Chapter 3 and the latent space

hypergraph model from Chapter 4 to the hyperbolic setting both present interest-

ing directions for future work. In the case of pairwise interactions, it is understood

empirically that many real world networks exhibit power law degree distributions

(Barabasi and Posfai (2016)) and it is interesting to explore whether an equivalent


behaviour can be found for interactions of arbitrary order. Adapting either of these

models to the hyperbolic setting will present many practical challenges. For example,

an approach for calculating the Cech complex with points in hyperbolic space must

be considered when adapting the methodology from Chapter 4. Finally, developing

an approach for posterior sampling in the Bayesian setting would also present an

important contribution to the literature.

Appendix A

Appendix for ‘Sequential Monte

Carlo and Dynamic Latent Space

Networks’

A.1 Gradient derivation for Euclidean distance and

binary observations

Here we derive the gradient for the DLSN model given in (3.3.1) - (3.3.5) for binary

data with a Euclidean metric. We wish to evaluate

∇Ut log p(Yt|Ut) = ∇Ut

∑i<j

{ηijtyijt − log(1 + eηijt)} (A.1.1)

Since it is not possible to consider ∇Ut , we focus on ∇Xkt so that each node is

moved conditional on all other nodes in a gradient nudge step.

∇Xit log p(Yt|Ut) =∑i<j

∇Xkt {ηijtyijt − log(1 + eηijt)} (A.1.2)

=∑i<j

∇Xkt

{(α− ‖Xit −Xjt‖)yijt − log(1 + eα−‖Xit−Xjt‖)

}(A.1.3)

121

APPENDIX A. SMC AND DYNAMIC LATENT SPACE NETWORKS 122

To evaluate (A.1.3) we require

∇Xit‖Xit −Xjt‖ =(Xit −Xjt)

‖Xit −Xjt‖(A.1.4)

and similarly

∇Xjt‖Xit −Xjt‖ = − (Xit −Xjt)

‖Xit −Xjt‖=

(Xjt −Xit)

‖Xit −Xjt‖(A.1.5)

Now, we find

∇Xkt{(α− ‖Xit −Xjt‖)yijt − log(1 + eα−‖Xit−Xjt‖)} (A.1.6)

=∑k<j

−ykjtXkt −Xjt

‖Xkt −Xjt‖−− Xkt −Xjt

‖Xkt −Xjt‖eα−‖Xkt−Xjt‖

1 + eα−‖Xkt−Xjt‖

+∑i<k

yiktXit −Xkt

‖Xit −Xkt‖−

Xit −Xkt

‖Xit −Xkt‖eα−‖Xit−Xkt‖

1 + eα−‖Xit−Xkt‖

(A.1.7)

=∑k<j

Xkt −Xjt

‖Xkt −Xjt‖

{eα−‖Xkt−Xjt‖

1 + eα−‖Xkt−Xjt‖− ykjt

}

−∑i<k

Xit −Xkt

‖Xit −Xkt‖

{eα−‖Xit−Xkt‖

1 + eα−‖Xit−Xkt‖− yikt

}(A.1.8)

=∑j 6=k

Xkt −Xjt

‖Xkt −Xjt‖



}(A.1.9)

where, in the last equality, we have used yijt = yjit, ‖Xit − Xjt‖ = ‖Xjt − Xit‖ and

(A.1.5).

Hence, we obtain the following expression for the gradient.

∇Xkt log p(Yt|Ut) =∑j 6=k

Xkt −Xjt

‖Xkt −Xjt‖



}(A.1.10)

Similar calculations can be made for a dot-product formulation.

A.2 Gradient derivation for parameter estimation

To estimate θ = (α, σ) using the scheme detailed in Section 3.4.2, we must find

expressions ford

dθlog p(Ut|Ut−1, θ) and

d

dθlog p(Yt|Ut, θ). For the model given in


(3.3.1) - (3.3.5), we obtain

∂

∂αlog p(Yt|Ut, θ) =

∂

∂α

∑i<j

{(α− ‖uit − ujt‖)yijt − log (1 + exp(α− ‖uit − ujt‖))}

(A.2.1)

=∑i<j

{yijt −

1

1 + exp(‖uit − ujt‖ − α)

}(A.2.2)

=∑i<j

(yijt − pijt) (A.2.3)

∂

∂σlog p(Yt|Ut, θ) = 0 (A.2.4)

∂

∂αlog p(Ut|Ut−1, θ) = 0 (A.2.5)

∂

∂σlog p(Ut|Ut−1, θ) =

∂

∂σ

N∑i=1

{d

2log(2π)− 1

2log(|σ2Id|)

−1

2(uit − ui,t−1)T (σ2Id)

−1(uit − ui,t−1)}(A.2.6)

=N∑i=1

{−dσ

+1

σ3(uit − ui,t−1)T (uit − ui,t−1)

}(A.2.7)

Since σ > 0, we opt to estimate log(σ). We obtain∂

∂ log(σ)log p(Ut|Ut−1, θ) by

multiplying (A.2.7) by σ. This follows from the chain rule.

A.3 Data simulation for Section 3.5.1

The data were simulated according to scenarios (S1), (S2) and (S3) by simulating the

latent trajectories according to

p(xt) = λp(xt−1) + (1− λ)gt (A.3.1)

where gt is the tth time step of a deterministic function. We specify these functions

so that they exhibit the behaviours described in scenarios (S1), (S2) and (S3).


0 20 40 60

1.0

1.4

Case 1

T

alph

a

0 20 40 60

1.0

1.4

Case 2

T

alph

a

0 20 40 60

1.0

1.4

Case 3

T

alph

a

5 10 15 20

1.0

1.4

nIts

alph

a

5 10 15 20

1.0

1.4

nIts

alph

a

5 10 15 20

1.0

1.4

nIts

alph

a



(a) α, online estimation.

0 20 40 60

1.0

1.4

Case 1

T

alph

a

0 20 40 60

1.0

1.4

Case 2

T

alph

a

0 20 40 60

1.0

1.4

Case 3

T

alph

a

5 10 15 20

1.0

1.4

nIts

alph

a

5 10 15 20

1.0

1.4

nIts

alph

a

5 10 15 20

1.0

1.4

nIts

alph

a



(b) α, offline estimation.

0 20 40 60

0.1

0.3

0.5

Case 1

T

sigm

a

0 20 40 60

0.1

0.3

0.5

Case 2

T

sigm

a

0 20 40 60

0.1

0.3

0.5

Case 3

T

sigm

a

5 10 15 20

0.1

0.3

0.5

nIts

sigm

a

5 10 15 20

0.1

0.3

0.5

nIts

sigm

a

5 10 15 20

0.1

0.3

0.5

nIts

sigm

a

(c) σ, online estimation.

0 20 40 60

0.1

0.3

0.5

Case 1

T

sigm

a

0 20 40 60

0.1

0.3

0.5

Case 2

T

sigm

a

0 20 40 60

0.1

0.3

0.5

Case 3

T

sigm

a

5 10 15 20

0.1

0.3

0.5

nIts

sigm

a

5 10 15 20

0.1

0.3

0.5

nIts

sigm

a

5 10 15 20

0.1

0.3

0.5

nIts

sigm

a

(d) σ, offline estimation.

Figure A.3.1: Summary of parameter estimates.

Appendix B

Appendix for ‘Latent Space

Modelling of Hypergraph Data’

B.1 Bookstein coordinates

In Bookstein coordinates a set of points are chosen as the anchor points. These

points are fixed in the space and all other points are translated, rotated and scaled

accordingly. In Appendix B.1.1 we describe the Bookstein coordinates in R2, and in

Appendix B.1.2 we describe the Bookstein coordinates in R3.

B.1.1 Bookstein coordinates in R2

In R2, we set the anchor points uB1 = (uB11, uB12) and uB2 = (uB21, u

B22) to be (−1/2, 0)

and (1/2, 0), respectively. Let UB denote the Bookstein coordinates and U denote

the untransformed coordinates. Then UB is given by

UB = cR(U − b)

=1√

(uB21 − uB11)2 + (uB22 − uB12)2

cos(a) sin(a)

− sin(a) cos(a)

U − 1

2

uB11 + uB21

uB12 + uB22

,

(B.1.1)

125

APPENDIX B. LS MODELLING OF HYPERGRAPH DATA 126

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●

●

−1 0 1 2

−2

−1

01

2

x

y

●

●

●

●

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●

−0.5 0.0 0.5 1.0

0.0

0.4

0.8

1.2

x

y

● ●

Figure B.1.1: Bookstein transformation in R2. Left: original coordinates. Right:

transformed Bookstein coordinates. The points highlighted in red are mapped to

(−1/2, 0) and (1/2, 0).

where a = arctan{(uB22 − uB12)/(uB21 − uB11)}. The Bookstein transformation can

hence be seen as a translation, rotation and rescaling of the coordinates U . Figure

B.1.1 shows an example of the Bookstein transformation in R2.

Furthermore, if U ∼ N (µ,Σ), then we know that UB ∼ N (µB,ΣB) where

µB = cR(µ− b), (B.1.2)

ΣB = c2RΣRT . (B.1.3)

B.1.2 Bookstein coordinates in R3

Section 4.3.3 of Dryden and Mardia (1998) gives the Bookstein transformation for U

where ui ∈ R3. Following from (B.1.1) we first set uB1 = (−1/2, 0, 0), uB2 = (1/2, 0, 0)

and uB3 = (uB31, uB32, 0). Then for i = 4, 5, . . . , N and l = 1, 2, 3 we calculate

wil = uil −1

2(uB1l + uB2l). (B.1.4)

The Bookstein coordinate uBi for i = 4, 5, . . . , N is then given by

uBi = R1R2R3(wi1, wi2, wi3)/D12 (B.1.5)


where

R1 =

1 0 0

0 cos(φ) sin(φ)

0 − sin(φ) cos(φ)

, R2 =

cos(ω) 0 sin(ω)

0 1 0

− sin(ω) 0 cos(ω)

(B.1.6)

R3 =

cos(θ) sin(θ) 0

− sin(θ) cos(θ) 0

0 0 1

, D12 = 2(w221 + w2

22 + w223)

1/2. (B.1.7)

Furthermore, we have

θ =arctan(w22/w21) (B.1.8)

ω =arctan(w23/(w221 + w2

22)1/2) (B.1.9)

φ =arctan

((w2

21 + w222)w33 − (w21w31 + w22w32)w23

(w221 + w2

22 + w223)

1/2(w21w32 − w31w22)

). (B.1.10)

We see that the transformation in R3 is more involved than in R2 since we need

to consider the effect of rotations over three different axes. Note that R1, R2 and R3

correspond to a rotation around the x, y and z axes, respectively.

B.2 Modifying the Hyperedge Indicators

In the generative model detailed in Algorithm 7, the indicators for all order k hyper-

edges are modified with probability ϕk. Since the probability ϕk is small, we expect

a small number of the(Nk

)possible order k hyperedges to be modified and so naively

simulating a Bernoulli(ϕk) random variable for each hyperedge is wasteful. Here we

discuss an alternative approach for this step which avoids considering all possible hy-

peredges, and we comment that this can easily be adapted for the model detailed in

Algorithm 8.

Instead of sampling(Nk

)Bernoulli random variables, we instead begin by sam-

pling the number of order k hyperedges whose indicator we modify, nk, from a


Binomial((Nk

),ϕk

). Then, we randomly sample nk hyperedges from EN,k. When sam-

pling a hyperedge, we want to sample an index from {i1 < i2 < · · · < ik|i1, i2, . . . , ik ∈

{1, 2, . . . , N}} and we do this by sampling i1 to ik in incrementally. This avoids

sampling from the(Nk

)possible combinations directly and so is more efficient.

We will now discuss this in more detail for hyperedges of order k = 2. To sample

indices i1 and i2 such that i1 < i2 we

1. Sample i1 with probability p(i1) =N − i1∑N−1i=1 (N − i)

, for i1 = 1, . . . , (N − 1).

2. Sample i2|i1 with probability p(i2|i1) =1

N − i1, for i2 = (i1 + 1), . . . , N .

A similar procedure can be used for arbitrary k.

Note that this procedure ignores the dependence between samples since, once a

hyperedge is sampled, the remaining hyperedges are sampled from a subset of hyper-

edges of size(Nk

)−1. However, we expect this effect to be negligible since the majority

of hyperedges are not modified.

B.3 Conditional Posterior Distributions

B.3.1 Conditional posterior for µ

The conditional posterior for µ is given by

p(µ|U ,Σ,mµ,Σµ) ∝ p(U |µ,Σ)p(µ|mµ,Σµ) (B.3.1)

where p(µ|mµ,Σµ) = N (mµ,Σµ) and p(U |µ,Σ) =∏N

i=1N (ui|µ,Σ).

We have

p(µ|U ,Σ,mµ,Σµ) ∝ exp

{−1

2

N∑i=1

(ui − µ)TΣ−1(ui − µ)− 1

2(µ−mµ)TΣ−1µ (µ−mµ)

}(B.3.2)


By recursively applying the result in Section of 8.1.7 Petersen and Pedersen (2012)

we obtain

p(µ|U ,Σ,mµ,Σµ) = N

((NΣ−1 + Σ−1µ )−1

(Σ−1

N∑i=1

ui + Σ−1µ mµ

), (NΣ−1 + Σ−1µ )−1

).

(B.3.3)

B.3.2 Conditional posterior for Σ

The conditional posterior for Σ is given by

p(Σ|U , µ,Φ, ν) ∝ p(U |µ,Σ)p(Σ|Φ, ν) (B.3.4)

where p(Σ|Φ, ν) =W−1(Φ, ν) and p(U |µ,Σ) =∏N

i=1N (ui|µ,Σ).

We have

p(Σ|U ,µ,Φ, ν)

∝N∏i=1

1

|Σ|1/2exp

{−1

2(ui − µ)tΣ−1(ui − µ)

}|Σ|−(ν+d+1)/2 exp

{−1

2tr(ΦΣ−1

)}(B.3.5)

∝ |Σ|−(ν+d+N+1)/2 exp

{−1

2

[tr

(Σ−1

N∑i=1

(ui − µ)(ui − µ)T

)+ tr(ΦΣ−1)

]}(B.3.6)

∝ |Σ|−(ν+d+N+1)/2 exp

{−1

2tr

([N∑i=1

(ui − µ)(ui − µ)T + Φ

]Σ−1

)}.

(B.3.7)

Line (B.3.7) follows from the symmetry of Σ and∑N

i=1(vi−µ)(vi−µ)T , and properties

of the trace operator.

Hence, we obtain

p(Σ|U , µ,Φ, ν) =W−1(

Φ +N∑i=1

(ui − µ)(ui − µ)T , ν +N

). (B.3.8)


B.3.3 Conditional posterior for ψ(0)k

The conditional posterior for ψ(0)k is given by

p(ψ

(0)k |U , r, hN,K , a

(0)k , b

(0)k

)∝ L

(U , r,ψ(1),ψ(0);hN,K

)p(ψ

(0)k |a

(0)k , b

(0)k

)(B.3.9)

where L(U , r,ψ(1),ψ(0);hN,K

)is as in (4.3.6) and p

(ψ

(0)k |a

(0)k , b

(0)k

)= Beta

(a(0)k , b

(0)k

).

We have

p(ψ(0)k |U , r, hN,K , a

(0)k , b

(0)k )

∝(ψ

(0)k

)d(01)k (gN,K(U ,r),hN,K) (1− ψ(0)

k

)d(00)k (gN,K(U ,r),hN,K) (ψ

(0)k

)a(0)k −1 (1− ψ(0)

k

)b(0)k −1(B.3.10)

∝(ψ

(0)k

)d(01)k (gN,K(U ,r),hN,K)+a(0)k −1

(1− ψ(0)

k

)d(00)k (gN,K(U ,r),hN,K)+b(0)k −1

. (B.3.11)

Hence, we obtain

p(ψ(0)k |U , r, hN,K , a

(0)k , b

(0)k )

= Beta(d(01)k (gN,K(U , r), hN,K) + a

(0)k , d

(00)k (gN,K(U , r), hN,K) + b

(0)k

).

(B.3.12)

B.3.4 Conditional posterior for ψ(1)k

The conditional posterior for ψ(1)k is given by

p(ψ

(1)k |U , r, hN,K , a

(1)k , b

(1)k

)∝ L

(U , r,ψ(1),ψ(0);hN,K

)p(ψ

(1)k |a

(1)k , b

(1)k

)(B.3.13)

where L(U , r,ψ(1),ψ(0);hN,K

)is as in (4.3.6) and p

(ψ

(1)k |a

(1)k , b

(1)k

)= Beta

(a(1)k , b

(1)k

).

We have

p(ψ(1)k |U , r, hN,K , a

(1)k , b

(1)k )

∝(ψ

(1)k

)d(10)k (gN,K(U ,r),hN,K) (1− ψ(1)

k

)d(11)k (gN,K(U ,r),hN,K) (ψ

(1)k

)a(1)k −1 (1− ψ(1)

k

)b(1)k −1(B.3.14)

∝(ψ

(1)k

)d(10)k (gN,K(U ,r),hN,K)+a(1)k −1

(1− ψ(1)

k

)d(11)k (gN,K(U ,r),hN,K)+b(1)k −1

. (B.3.15)


Hence, we obtain

p(ψ(1)k |U , r, hN,K , a

(1)k , b

(1)k )

= Beta(d(10)k (gN,K(U , r), hN,K) + a

(1)k , d

(11)k (gN,K(U , r), hN,K) + b

(1)k

).

(B.3.16)

B.4 MCMC initialisation

For the MCMC scheme in Algorithm 9, a random initialisation is likely to perform

poorly. Here we discuss our approach for initialising the MCMC scheme, and we begin

with the the latent coordinates U .

In Sarkar and Moore (2006), the authors present a procedure for inferring the

latent coordinates in the scenario where the network is temporally evolving. Their

scheme begins with an initialisation which uses generalised multidimensional scaling

(GMDS). Traditional MDS (see Cox and Cox (2000)) finds a set of coordinates in Rd

whose pairwise distances correspond to a distance matrix specified as an input. In

GMDS, the distance is extended beyond the Euclidean distance and, in our context,

we use the shortest path between nodes i and j as the distance measure. To calcu-

late the shortest paths we introduce a weighted adjacency matrix which incorporates

the intuition than nodes which appear in a hyperedge are likely closer than nodes

which are only connected by a pairwise edge. Once we have an initial value of U

we then transform these coordinates onto the Bookstein space of coordinates. Our

initialisation procedure for U is given in Algorithm 10.

The radii r depend on the scale of U , and so they are initialised in terms of

U0. Given the initial latent coordinates, r0 is chosen to be the minimum radius

which induces all edges that are present in hN,K . The noise parameters ψ(0) and ψ(1)

are initialised by sampling from their prior, where the prior values suggest that the

perturbations are small.

To initialise the parameters µ and Σ we use an ABC scheme (see Marin et al.

(2012) for an overview). In this scheme we first sample µ and Σ from their priors.


Algorithm 10 Initialise U .

Input: Observed hypergraph hN,K

1) Let A ∈ RN×N denote a weighted adjacency matrix.

For i, j ∈ {1, 2, . . . , N}, if {i, j} are connected by a hyperedge

- let A(i,j) = 1 if {i, j} are only connected by a hyperedge of order k = 2,

- let A(i,j) = λ if {i, j} are connected by a hyperedge of order k > 2.

2) Find the distance matrix D ∈ RN×N , where D(i,j) is the shortest path between

nodes {i, j} in the weighted graph determined by A. For i = j, let D(i,j) = 0.

3) Apply MDS to D to obtain coordinates U0 ∈ RN×d.

4) Specify the index of the anchor points, and transform U0 onto Bookstein

coordinates (see Appendix B.1).

Conditional on these samples, we then sample a hypergraph. By comparing summary

statistics of the sampled hypergraph with the observed hypergraph, we determine

whether or not our sampled hypergraph is similar enough to the observed hypergraph.

If so, we accept the sampled µ and Σ. We choose the number of hyperedges of order

k = 2, the number of hyperedges of order k = 3 and the number of triangles as our

summary statistics. This initialisation scheme is detailed in Algorithm 11 and the full

initialisation scheme is given in Algorithm 12.

B.5 Practicalities

To implement the MCMC scheme given in Algorithm 9 there are a number of practical

considerations we must address. In this section we comment on these where, in B.5.1

we discuss an approach for determining the presence of a hyperedge of arbitrary order

and in B.5.2 we discuss efficient evaluation of the likelihood.


Algorithm 11 Initialise µ and Σ.

Input: Hypergraph hN,K , r0, ψ(0)0 , ψ

(1)0 , µ ∼ N (mµ,Σµ), Σ ∼ W−1(Φ, ν), Nsmp

and ε.

1) Calculate T (hN,K), where T (·) is a vector of hypergraph summary statistics.

Let n = 0.

2) While n < Nsmp

-Sample µ∗ ∼ N (mµ,Σµ) and Σ∗ ∼ W−1(Φ, ν).

-Sample u∗i ∼ N (µ∗,Σ∗) for i = 1, 2, . . . , N .

Let U ∗ be the N × d matrix whose ith row is u∗i .

-Given initial r0, determine the hypergraph gN,K(U ∗, r0).

-Let g∗N,K by the hypergraph obtained by modifying gN,K(U ∗, r0) with noise

ψ(0)0 and ψ

(1)0

-Calculate T(g∗N,K

).

-If |T (hN,K)− T (g∗N,K)| < ε

Accept samples µ∗ and Σ∗.

3) Let µ0 and Σ0 be the average of Nsmp samples.

B.5.1 Smallest Enclosing Ball

Here we discuss an approach for determining the presence of a hyperedge conditional

on U and r. Recall that a hyperedge ek = {i1, i2, . . . , ik} is present if Brk(ui1) ∩

Brk(ui2) ∩ · · · ∩ Brk(uik) 6= ∅. Hence, in order to determine whether yek = 1, we

must find whether or not the sets corresponding to the nodes in the hyperedge have

a non-empty intersection.

Alternatively, note that it is equivalent to determine whether the coordinates

{ui1 , ui2 , . . . , uik} are contained within a ball of radius rk (see section 3.2 of Edels-

brunner and Harer (2010)). Figure B.5.1 shows this for an example with k = 3. This

means that we can rephrase the above into the following procedure.

1. Determine the smallest enclosing ball B for the coordinates {ui1 , ui2 , . . . , uik}.


Algorithm 12 Procedure for initialising MCMC scheme in Algorithm 9.

Input Observed hypergraph hN,K

1) Determine U0 by applying Algorithm 10.

2) Let initial radii r0 be the smallest radii which induce all hyperedges observed in

hN,K , conditional on U0.

3) Sample ψ(0)0 and ψ

(1)0 from their prior distributions.

4) Sample µ0 and Σ0 by applying Algorithm 11.

●

●

●

●

● r∗

Figure B.5.1: The blue shaded regions correspond to Br(ui), for i = 1, 2, 3, and the

purple shaded region is the smallest enclosing ball of the points. The statements

r∗ < r and Br(u1) ∩Br(u2) ∩Br(u3) 6= ∅ are equivalent.

2. If the radius of B is less than rk, the hyperedge ek = {i1, i2, . . . , ik} is present

in the hypergraph.

To compute the smallest enclosing ball we can rely on the the miniball algorithm

(see Section 3.2 of Edelsbrunner and Harer (2010)), which may be also be referred to

as the minidisk algorithm (see Section 4.7 of Berg et al. (2008)). Before providing the

algorithmic details, we will first discuss the intuition behind this algorithm. In the

discussion below, we will follow the explanation of Edelsbrunner and Harer (2010).

For a set of points, it is clear that a given point is either contained within B or

it lies on the boundary of B. The set of points on the boundary entirely determine

B and, when the number of points is much larger than the dimension, the chance


of a point lying on the boundary is small. Miniball exploits these facts to partition

the set of points into those which are contained within B and those which lie on the

boundary. The algorithm begins by sampling a point u from the full set of points

ui1...ik = {ui1 , ui2 , . . . , uik}. If u lies within the smallest enclosing ball of ui1...ik \ u,

then we know it lies within B and so it does not influence B. Alternatively, if u lies

on the boundary then we must include it in the set which determines B. Miniball

iterates over all points in this way to determine the set of points on the boundary.

Then, once we have the minimum closing ball B with radius r∗, we check whether

r∗ < rk to determine the presence of a hyperedge.

To determine the set of order k hyperedges present in the graph, we rely on the

simplicial property of the Cech complex (see Section 4.2.3). By observing that all

subsets of an order k hyperedge must also be connected, we reduce the search space

from all possible hyperedges to those whose subsets are present.

We now present the algorithmic details of the miniball algorithm (see Section 3.2

of Edelsbrunner and Harer (2010)). For the edge ek = {i1, i2, . . . , ik}, we let σ1 ⊆ ek

and σ2 ⊆ ek denote subsets which partition ek so that σ1 ∩ σ2 = ek. After a pass of

the algorithm, the set σ2 will contain the index of nodes which lie on the boundary of

the smallest enclosing ball B. Hence, σ2 represents the nodes which determine B. We

initialise the miniball algorithm with σ1 = ek and σ2, and iteratively identify which

nodes from σ1 belong in σ2. The procedure is given in Algorithm 13.

An alternative description of this algorithm can be found in Section 4.7 of Berg

et al. (2008), and for efficient implementation of the Cech complex we rely on the

GUDHI C++ library (The GUDHI Project (2015)).

B.5.2 Evaluating L(U , r,ψ(1),ψ(0);hN,K)

The likelihood given in (4.3.6) requires the enumeration of hyperedge discrepancies

between the observed hypergraph hN,K and the induced hypergraph gN,K(U , r). In

this section we note that this does not require a summation over all possible hyper-


Algorithm 13 Miniball

1) Set σ1 = ek and σ2 = ∅

2) if σ1 = ∅, compute the miniball B of σ2

else choose u ∈ σ1

-Calculate the miniball B which contains the points σ1 \ u in its interior and

the

points σ2 on its boundary

-if u /∈ B, then set B to be the miniball B which contains the points σ1 \ u

in its interior and the points σ2 ∪ u on its boundary

edges, and so can be computed far more efficiently than (4.3.5) suggests. We first

discuss evaluation of (4.3.6), and then observe that (4.3.3) can be evaluated in a

similar way.

To evaluate the likelihood we have the hyperedges present in hN,K and the hy-

peredges present in gN,K(U , r). In practice, as the data examples from Section 4.8

suggest, the number of hyperedges in each of these hypergraphs is much smaller than

the number of possible hyperedges∑K

k=2

(Nk

). Let n

(h)k and n

(g)k denote the number of

order k hyperedges in hN,K and gN,K(U , r), respectively. To evaluate the likelihood,

we begin by enumerating the number of order k hyperedges which are present in both

hypergraphs to obtain d(11)k (gN,K(U , r), hN,K). This can easily be computed by eval-

uating the number of intersection between the hyperedges in hN,K and gN,K(U , r).

Then, for k = 2, 3, . . . , K, it follows that

d(10)k (gN,K(U , r), hN,K) = n

(g)k − d

(11)k (gN,K(U , r), hN,K), (B.5.1)

d(01)k (gN,K(U , r), hN,K) = n

(h)k − d

(11)k (gN,K(U , r), hN,K), (B.5.2)

d(00)k (gN,K(U , r), hN,K) =

(N

k

)−[d(11)k (gN,K(U , r), hN,K)

+d(10)k (gN,K(U , r), hN,K) + d

(01)k (gN,K(U , r), hN,K)

].

(B.5.3)

Hence, we are able to avoid summation over all possible hyperedges. We can easily


calculate the distance specified in (4.3.2) from the above, by observing that it is given

by the sum of (B.5.1) and (B.5.2).

B.6 Proofs for Section 4.6

B.6.1 Proof of Proposition 4.6.1

We have Ui ∼ N (µ,Σ) for i = 1, 2, . . . , N and Σ = diag(σ21, σ

22, . . . , σ

2d). Our goal is

to obtain an expression for pe2 , and we begin by noting

pe2 = P (‖Ui − Uj‖ ≤ 2r2|µ,Σ) = P((Ui − Uj)T (Ui − Uj) ≤ 4(r2)

2|µ,Σ). (B.6.1)

Hence, we consider the distribution of XTijXij where Xij = Ui − Uj.

From properties of the Normal distribution, we have that Xij = Ui−Uj ∼ N (0, 2Σ)

and, from Section 1 of Duchesne and Micheaux (2010), we find

XTijXij|Σ =

d∑l=1

λlχ21 (B.6.2)

where λl is the lth eigenvalue of 2Σ. Since Σ is diagonal, the eigenvalues of 2Σ are

given by λl = 2σ2l . Furthermore, since a χ2

1 distribution is equivalent to a Γ(1/2, 2)

distribution, we have

Zij|Σ = XTijXij|Σ ∼

d∑l=1

Γ

(1

2, 2(2σ2

l )

). (B.6.3)

Hence, we have the result.

B.6.2 Proof of Lemma 4.6.1

Recall that y(g)ek = 1 and y

(g∗)ek = 1 indicate that the hyperedge ek is present in g and

in g∗, respectively. To begin we consider the probability of ek being present in g∗. We

may observe y(g∗)ek = 1 from either y

(g)ek = 1 or y

(g)ek = 0. In the first case, we want to

keep the state of ek unaltered and, in the second case, we want to modify the state of


the edge. Hence we have

P (yek = 1(g∗)|Σ) = (1−ϕk)P (y(g)ek= 1|Σ) + ϕk(1− P (y(g)ek

= 1|Σ)). (B.6.4)

The degree of the ith node with respect to order k hyperedges is obtained by

summing over all possible hyperedges ek that are incident to i. We have

Degg(i,k) =∑

{ek∈EN,k|i∈ek}

y(g)ek(B.6.5)

and, in total, there are(N−1k−1

)possible order k hyperedges that are incident to i.

Therefore, it follows that

E[Degg(i,k)|Σ

]=

(N − 1

k − 1

)P (yek = 1(g)|Σ). (B.6.6)

Note that (B.6.6) is valid for dependent probabilities and we rely on this for hyperedges

of order k ≥ 3. As an example, consider the hyperedges {i, j, k} and {i, j, l} when

k = 3. It is clear that both of these hyperedges depend on both i and j, and so

there is dependence between the hyperedges and the summation in the calculation of

Degg(i,k).

Now we consider E[Degg

∗

(i,k)

]. By the law of total expectation, we have

E[Degg

∗

(i,k)|ϕk,Σ]

= E[E[Degg

∗

(i,k)|Degg(i,k)

]|ϕk,Σ

](B.6.7)

From (B.6.4), it follows that

E[Degg

∗

(i,k)|ϕk,Σ]

= E

[(1−ϕk)Degg(i,k) + ϕk

((N − 1

k − 1

)−Degg(i,k)

)|ϕk,Σ

](B.6.8)

= (1−ϕk)E[Degg(i,k)|ϕk,Σ

]+ ϕk

((N − 1

k − 1

)− E

[Degg(i,k)|ϕk,Σ

]).

(B.6.9)

Using (B.6.5), we obtain

E[Degg

∗

(i,k)|ϕk, µ,Σ]

=

(N − 1

k − 1

)[(1−ϕk)P (yek = 1(g)|µ,Σ) + ϕk(1− P (y(g)ek

= 1|µ,Σ))].

(B.6.10)

The final result then follows directly from Observation (O2).


B.6.3 Proof of Theorem 4.6.1

To begin we will consider the degree distribution of hyperedges of order k = 2. From

(B.6.4) we have that the probability of an edge e2 being present in g∗ is given by

P(y(g

∗)e2

= 1|Σ, r2,ϕ2

)= (1−ϕ2)pe2 + ϕ2(1− pe2), (B.6.11)

where pe2 is the probability that e2 is present in g.

The degree distribution of the ith node is a sum of independent Bernoulli trials

since the edges {i, j} and {i, k} occur independently given conditioning on i. There

are N − 1 possible order 2 edges which contain i. Hence it follows that

Degg∗

(i,2)|r2,ϕ2,Σ ∼ Binomial (N − 1, (1−ϕ2)pe2 + ϕ2(1− pe2)) . (B.6.12)

The degree of the ith node for order k hyperedges in g∗ is given by

Degg∗

(i,k) =∑

{ek∈EN,k|i∈ek}

yg∗

ek. (B.6.13)

Now we find the degree distribution for hyperedges of order k = 3. Consider the

hyperedges {i, j, k} and {i, j, l} and note that these hyperedges both depend on i

and j. It is clear that P (y(g)ijk = 1) and P (y

(g)ijl = 1) are not independent and so the

argument used for the degree distribution involving hyperedges of order k = 2 is no

longer appropriate. Now we need to consider the sum of(N−12

)dependent Bernoulli

trials. When pijk small, we can approximate this by a Poisson distribution with rate

parameter λ =∑{ek∈EN,3|i∈ek} y

(g∗)ek . However, we note that this result can be extended

using results presented in Teerapabolarn (2014)).

B.7 Prior and Posterior Predictive Degree Distri-

butions

Here we present additional plots for the simulation study detailed in Section 4.7.2.

Figure B.7.1 compares the predictives for the connections between the observed nodes


and the newly simulated nodes, and Figure B.7.2 compares the predictives for con-

nections occurring among the newly simulated nodes only. In both figures, we see a

close correspondence between the prior and predictive distributions.

0 2 4 6 8 10 12 14 16 18

Degree

Pro

babi

lity

0.00

0.05

0.10

0.15

0.20

●●●●●●●●●●●●

●●

●

●

●

●●

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.10

0.20

'True' Degree Distribution

Est

imat

ed D

egre

e D

istr

ibut

ion


0 21 45 69 93 120 151 182 213

Degree

Pro

babi

lity

0.00

0.02

0.04

0.06

0.08

●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●●

●●●●●●●●●●●

●●●●●●●●●●●

●●●●●●●●●

●

●

●

0.00 0.02 0.04 0.06 0.08

0.00

0.02

0.04

0.06

0.08


Est

imat

ed D

egre

e D

istr

ibut

ion


Figure B.7.1: Comparison of prior and posterior predictive degree distributions for

hyperedges occurring between theN∗ = 10 newly simulated nodes and the nodes of the

observed hypergraph. In each figure, the left panel shows the prior predictive degree

distribution, and the right panel shows a qq-plot of the prior and posterior predictive

degree distributions. Figures B.7.1a and B.7.1b show the degree distributions for

hyperedges of order 2 and 3, respectively.


0 1 2 3 4 5 6 7 8 9

Degree

Pro

babi

lity

0.00

0.10

0.20

●●●●

●

●

●

● ●

0.00 0.10 0.20 0.30

0.00

0.10

0.20

0.30


Est

imat

ed D

egre

e D

istr

ibut

ion


0 3 6 9 12 16 20 24 28

Degree

Pro

babi

lity

0.00

0.10

0.20

0.30

●●●●●●●●●●●●●●●●●●●●●●●

●●●

●

●

●

●

0.0 0.1 0.2 0.3

0.0

0.1

0.2

0.3


Est

imat

ed D

egre

e D

istr

ibut

ion


Figure B.7.2: Comparison of prior and posterior predictive degree distributions for

hyperedges occurring between the N∗ = 10 newly simulated nodes only. In each figure,

the left panel shows the prior predictive degree distribution, and the right panel shows

a qq-plot of the prior and posterior predictive degree distributions. Figures B.7.2a

and B.7.2b show the degree distributions for hyperedges of order 2 and 3, respectively.

Bibliography

Airoldi, E. M., Blei, D. M., Fienberg, S. E., and Xing, E. P. (2009). Mixed membership

stochastic blockmodels. In Koller, D., Schuurmans, D., Bengio, Y., and Bottou,

L., editors, Advances in Neural Information Processing Systems 21, pages 33–40.

Curran Associates, Inc.

Aksoy, S., Kolda, T. G., and Pinar, A. (2016). Measuring and Modeling Bipartite

Graphs with Community Structure. arXiv e-prints, page arXiv:1607.08673.

Akyildiz, O. D. and Mıguez, J. (2019). Nudging the particle filter. Statistics and

Computing.

Aldous, D. J. (1981). Representations for partially exchangeable arrays of random

variables. Journal of Multivariate Analysis, 11(4):581 – 598.

Aminikhanghahi, S. and Cook, D. J. (2017). A survey of methods for time series

change point detection. Knowledge and Information Systems, 51(2):339–367.

Andrieu, C., Doucet, A., and Holenstein, R. (2010). Particle markov chain monte carlo

methods. Journal of the Royal Statistical Society: Series B (Statistical Methodol-

ogy), 72(3):269–342.

Athreya, A., Fishkind, D. E., Tang, M., Priebe, C. E., Park, Y., Vogelstein, J. T.,

Levin, K., Lyzinski, V., Qin, Y., and Sussman, D. L. (2017). Statistical inference

on random dot product graphs: a survey. Journale of Machine Learning Research,

18:226:1–226:92.

142

BIBLIOGRAPHY 143

Avella-Medina, M., Parise, F., Schaub, M., and Segarra, S. (2018). Centrality mea-

sures for graphons: accounting for uncertainty in networks. IEEE Transactions on

Network Science and Engineering.

Barabasi, A.-L. and Albert, R. (1999). Emergence of scaling in random networks.

Science, 286(5439):509–512.

Barabasi, A.-L. and Posfai, M. (2016). Network science. Cambridge University Press,

Cambridge.

Bengtsson, T., Bickel, P., Li, B., et al. (2008). Curse-of-dimensionality revisited:

Collapse of the particle filter in very large scale systems. In Probability and statistics:

Essays in honor of David A. Freedman, pages 316–334. Institute of Mathematical

Statistics.

Benson, A. R., Abebe, R., Schaub, M. T., Jadbabaie, A., and Kleinber, J. (2018).

Simplicial closure and higher-order link prediction. Proceedings of the National

Academy of Sciences, 115(48):E11221–E11230.

Berg, M. d., Cheong, O., Kreveld, M. v., and Overmars, M. (2008). Computational

Geometry: Algorithms and Applications. Springer-Verlag TELOS, Santa Clara, CA,

USA, 3rd ed. edition.

Beskos, A., Crisan, D., and Jasra, A. (2014a). On the stability of sequential monte

carlo methods in high dimensions. The Annals of Applied Probability, 24(4):1396–

1445.

Beskos, A., Crisan, D., Jasra, A., Kamatani, K., and Zhou, Y. (2017). A stable

particle filter for a class of high-dimensional state-space models. Advances in Applied

Probability, 49(1):2448.

Beskos, A., Crisan, D. O., Jasra, A., and Whiteley, N. (2014b). Error bounds and

BIBLIOGRAPHY 144

normalising constants for sequential monte carlo samplers in high dimensions. Ad-

vances in Applied Probability, 46(1):279–306.

Bickel, P., Li, B., and Bengtsson, T. (2008). Sharp failure rates for the bootstrap

particle filter in high dimensions, volume Volume 3 of Collections, pages 318–329.

Institute of Mathematical Statistics, Beachwood, Ohio, USA.

Biswal, B. B., Mennes, M., Zuo, X.-N., Gohel, S., Kelly, C., Smith, S. M., Beckmann,

C. F., Adelstein, J. S., Buckner, R. L., Colcombe, S., Dogonowski, A.-M., Ernst,

M., Fair, D., Hampson, M., Hoptman, M. J., Hyde, J. S., Kiviniemi, V. J., Kotter,

R., Li, S.-J., Lin, C.-P., Lowe, M. J., Mackay, C., Madden, D. J., Madsen, K. H.,

Margulies, D. S., Mayberg, H. S., McMahon, K., Monk, C. S., Mostofsky, S. H.,

Nagel, B. J., Pekar, J. J., Peltier, S. J., Petersen, S. E., Riedl, V., Rombouts,

S. A. R. B., Rypma, B., Schlaggar, B. L., Schmidt, S., Seidler, R. D., Siegle,

G. J., Sorg, C., Teng, G.-J., Veijola, J., Villringer, A., Walter, M., Wang, L.,

Weng, X.-C., Whitfield-Gabrieli, S., Williamson, P., Windischberger, C., Zang, Y.-

F., Zhang, H.-Y., Castellanos, F. X., and Milham, M. P. (2010). Toward discovery

science of human brain function. Proceedings of the National Academy of Sciences,

107(10):4734–4739.

Bloem-Reddy, B. and Orbanz, P. (2018). Random-walk models of network formation

and sequential monte carlo methods for graphs. Journal of the Royal Statistical

Society: Series B (Statistical Methodology), 80(5):871–898.

Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions.

Statist. Sci., 1(2):181–222.

Borg, I. and Groenen, P. (1997). Modern Multidimensional Scaling: Theory and

Applications. Springer series in statistics. Springer.

Borgs, C. and Chayes, J. (2017). Graphons: A nonparametric method to model,

estimate, and design algorithms for massive networks. In Proceedings of the 2017

BIBLIOGRAPHY 145

ACM Conference on Economics and Computation, EC ’17, pages 665–672, New

York, NY, USA. ACM.

Brockwell, A., Del Moral, P., and Doucet, A. (2012). Sequentially interacting Markov

chain Monte Carlo methods. ArXiv e-prints.

Cai, D., Campbell, T., and Broderick, T. (2016). Edge-exchangeable graphs and

sparsity. In Lee, D. D., Sugiyama, M., Luxburg, U. V., Guyon, I., and Garnett, R.,

editors, Advances in Neural Information Processing Systems 29, pages 4249–4257.

Curran Associates, Inc.

Campbell, T., Cai, D., and Broderick, T. (2018). Exchangeable trait allocations.

Electron. J. Statist., 12(2):2290–2322.

Cappe, O. (2011). Online em algorithm for hidden markov models. Journal of Com-

putational and Graphical Statistics, 20(3):728–749.

Caron, F. and Fox, E. B. (2017). Sparse graphs using exchangeable random mea-

sures. Journal of the Royal Statistical Society: Series B (Statistical Methodology),

79(5):1295–1366.

Carpenter, J., P.Clifford, and Fearnhead, P. (1999). Improved particle filter for non-

linear problems. IEE Proceedings - Radar, Sonar and Navigation, 146:2–7(5).

Carvalho, C. M., Johannes, M. S., Lopes, H. F., and Polson, N. G. (2010). Particle

learning and smoothing. Statistical Science, 25(1):88–106.

Chien, I., Lin, C.-Y., and Wang, I.-H. (2018). Community detection in hypergraphs:

Optimal statistical limit and efficient algorithms. In Storkey, A. and Perez-Cruz,

F., editors, Proceedings of the Twenty-First International Conference on Artificial

Intelligence and Statistics, volume 84 of Proceedings of Machine Learning Research,

pages 871–879, Playa Blanca, Lanzarote, Canary Islands. PMLR.

BIBLIOGRAPHY 146

Chopin, N., Jacob, P. E., and Papaspiliopoulos, O. (2013). Smc2: an efficient algo-

rithm for sequential analysis of state space models. Journal of the Royal Statistical

Society: Series B (Statistical Methodology), 75(3):397–426.

Cooley, O., Fang, W., Del Giudice, N., and Kang, M. (2018). Subcritical ran-

dom hypergraphs, high-order components, and hypertrees. arXiv e-prints, page

arXiv:1810.08107.

Coulson, M., Gaunt, R., and Reinert, G. (2016). Poisson approximation of subgraph

counts in stochastic block models and a graphon model. ESAIM: Probability and

Statistics.

Cox, T. F. and Cox, M. A. A. (2000). Multidimensional Scaling. Chapman & Hall, 2

edition.

Crane, H. and Dempsey, W. (2018). Edge exchangeable models for interaction net-

works. Journal of the American Statistical Association, 113(523):1311–1326.

Csardi, G. and Nepusz, T. (2006). The igraph software package for complex network

research. InterJournal, Complex Systems:1695.

D’Angelo, S., Alfo, M., and Murphy, T. B. (2018). Modelling heterogeneity in latent

space models for multidimensional networks.

D’Angelo, S., Murphy, T. B., and Alfo, M. (2019). Latent space modelling of multi-

dimensional networks with application to the exchange of votes in eurovision song

contest. Ann. Appl. Stat., 13(2):900–930.

De Ridder, S., Vandermarliere, B., and Ryckebusch, J. (2016). Detection and localiza-

tion of change points in temporal networks with the aid of stochastic block models.

Journal of Statistical Mechanics: Theory and Experiment, 2016(11):113302.

Del Moral, P. and Murray, L. M. (2015). Sequential monte carlo with highly informa-

tive observations. SIAM/ASA Journal on Uncertainty Quantification, 3(1):969–997.

BIBLIOGRAPHY 147

Delaunay, B. (1934). Sur la sphere vide. Bulletin de l’Academie des Sciences de

l’URSS, Classe des Sciences Mathematiques et Naturelles, pages 793–800.

Dempsey, W., Oselio, B., and Hero, A. (2019). Hierarchical network models for struc-

tured exchangeable interaction processes. arXiv e-prints, page arXiv:1901.09982.

Douc, R. and Cappe, O. (2005). Comparison of resampling schemes for particle

filtering. In ISPA 2005. Proceedings of the 4th International Symposium on Image

and Signal Processing and Analysis, 2005., pages 64–69.

Doucet, A., De Freitas, N., and Gordon, N., editors (2001). Sequential Monte Carlo

methods in practice.

Doucet, A., Godsill, S., and Andrieu, C. (2000). On sequential monte carlo sampling

methods for bayesian filtering. Statistics and Computing, 10(3):197–208.

Doucet, A. and Johansen, A. (2008). A tutorial on particle filtering and smoothing:

Fifteen years later. Technical report.

Dryden, I. L. and Mardia, K. V. (1998). Statistical Shape Analysis. Wiley, Chichester.

Duchesne, P. and Micheaux, P. L. D. (2010). Computing the distribution of quadratic

forms: Further comparisons between the liutangzhang approximation and exact

methods. Computational Statistics and Data Analysis, 54(4):858 – 862.

Durante, D. and Dunson, D. B. (2014). Nonparametric Bayes dynamic modelling of

relational data. Biometrika, 101(4):883–898.

Durante, D., Dunson, D. B., and Vogelstein, J. T. (2016). Nonparametric bayes mod-

eling of populations of networks. Journal of the American Statistical Association.

Durante, D., Mukherjee, N., and Steorts, R. C. (2017). Bayesian learning of dynamic

multilayer networks. J. Mach. Learn. Res., 18(1):1414–1442.

BIBLIOGRAPHY 148

Edelsbrunner, H. and Harer, J. (2010). Computational Topology - an Introduction.

American Mathematical Society.

Edelsbrunner, H., Kirkpatrick, D., and Seidel, R. (1983). On the shape of a set of

points in the plane. IEEE Trans. Inf. Theor., 29(4):551–559.

Eldridge, J., Belkin, M., and Wang, Y. (2016). Graphons, mergeons, and so on! In

Proceedings of the 30th International Conference on Neural Information Processing

Systems, NIPS’16, pages 2315–2323, USA. Curran Associates Inc.

Elie de Panafieu (2015). Phase transition of random non-uniform hypergraphs. Jour-

nal of Discrete Algorithms, 31:26 – 39. 24th International Workshop on Combina-

torial Algorithms (IWOCA 2013).

Elvira, V., Martino, L., and Robert, C. P. (2018). Rethinking the Effective Sample

Size. ArXiv e-prints.

Erdos, P. and Renyi, A. (1959). On random graphs, I. Publicationes Mathematicae

(Debrecen), 6:290–297.

Fasano, A., Rebaudo, G., Durante, D., and Petrone, S. (2019). A Closed-Form Filter

for Binary Time Series. arXiv e-prints, page arXiv:1902.06994.

Fearnhead, P. (2002). Markov chain monte carlo, sufficient statistics and particle

filters. Journal of Computational and Graphical Statistics, 11(4):848–862.

Fosdick, B. K., McCormick, T. H., Murphy, T. B., Ng, T. L. J., and Westling, T.

(2019). Multiresolution network models. Journal of Computational and Graphical

Statistics, 28(1):185–196.

Frank, O. (1991). Statistical analysis of change in networks. Statistica Neerlandica,

45(3):283–293.

Frank, O. and Strauss, D. (1986). Markov graphs. Journal of the American Statistical

Association, 81(395):832–842.

BIBLIOGRAPHY 149

Friel, N., Rastelli, R., Wyse, J., and Raftery, A. E. (2016). Interlocking directorates

in irish companies using a latent space model for bipartite networks. Proceedings

of the National Academy of Sciences, 113(24):6629–6634.

Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo: Stochastic

Simulation for Bayesian Inference. Chapman and Hall/CRC, 2 edition.

Gao, M. and Zhang, H. (2012). Sequential monte carlo methods for parameter esti-

mation in nonlinear state-space models. Computers and Geosciences, 44:70 – 77.

Gemmetto, V., Barrat, A., and Cattuto, C. (2014). Mitigation of infectious disease

at school: targeted class closure vs school closure. In BMC Infectious Diseases.

Ghoshdastidar, D. and Dukkipati, A. (2014). Consistency of spectral partitioning of

uniform hypergraphs under planted partition model. In Ghahramani, Z., Welling,

M., Cortes, C., Lawrence, N. D., and Weinberger, K. Q., editors, Advances in

Neural Information Processing Systems 27, pages 397–405. Curran Associates, Inc.

Gilbert, E. N. (1959). Random graphs. Ann. Math. Statist., 30(4):1141–1144.

Gilks, W. R. and Berzuini, C. (2001). Following a moving targetmonte carlo inference

for dynamic bayesian models. Journal of the Royal Statistical Society: Series B

(Statistical Methodology), 63(1):127–146.

Gilliland, D. C. (1962). Integral of the bivariate normal distribution over an offset

circle. Journal of the American Statistical Association, 57(300):758–768.

Godsill, S. and Clapp, T. (2001). Improvement Strategies for Monte Carlo Particle

Filters, pages 139–158. Springer New York, New York, NY.

Goldenberg, A., Zheng, A. X., Fienberg, S. E., and Airoldi, E. M. (2010). A survey

of statistical network models. Found. Trends Mach. Learn., 2(2):129–233.

Gollini, I. and Murphy, T. B. (2016). Joint modeling of multiple network views.

Journal of Computational and Graphical Statistics, 25(1):246–265.

BIBLIOGRAPHY 150

Goodman, L. A. (1974). Exploratory latent structure analysis using both identifiable

and unidentifiable models. Biometrika, 61(2):215–231.

Gordon, N., Salmond, D., and Smith, A. (1993). Novel approach to nonlinear/non-

gaussian bayesian state estimation. IEEE Proceedings F, Radar and Signal Pro-

cessing, 140(2):107–113.

Gormley, I. C. and Murphy, T. B. (2010). A mixture of experts latent position cluster

model for social network data. Statistical Methodology, 7(3):385 – 405. Special Issue

on Statistical Methods for the Social Sciences. Honoring the 10th Anniversary of

the Center for Statistics and the Social Sciences at the University of Washington.

Gromov, M. (1987). Hyperbolic groups. Essays in Group Theory, Mathematical

Sciences Research Institute Publications.

Gunawan, D., Dang, K.-D., Quiroz, M., Kohn, R., and Tran, M.-N. (2018). Subsam-

pling sequential monte carlo for static bayesian models.

Handcock, M. S., Raftery, A. E., and Tantrum, J. M. (2007). Model-based clustering

for social networks. Journal of the Royal Statistical Society: Series A (Statistics in

Society), 170(2):301–354.

Heard, N. A. and Turcotte, M. J. M. (2017). Adaptive sequential monte carlo for

multiple changepoint analysis. Journal of Computational and Graphical Statistics,

26(2):414–423.

Hoff, P. (2008a). Modeling homophily and stochastic equivalence in symmetric re-

lational data. In Platt, J. C., Koller, D., Singer, Y., and Roweis, S. T., editors,

Advances in Neural Information Processing Systems 20, pages 657–664. Curran

Associates, Inc.

Hoff, P. D. (2008b). Multiplicative latent factor models for description and predic-

BIBLIOGRAPHY 151

tion of social networks. Computational and Mathematical Organization Theory,

15(4):261.

Hoff, P. D. (2011). Hierarchical multilinear models for multiway data. Computational

Statistics & Data Analysis, 55(1):530–543.

Hoff, P. D., Raftery, A. E., and Handcock, M. S. (2002). Latent space ap-

proaches to social network analysis. Journal of the American Statistical Association,

97(460):1090–1098.

Holland, P. W., Laskey, K. B., and Leinhardt, S. (1983). Stochastic blockmodels:

First steps. Social Networks, 5(2):109 – 137.

Holland, P. W. and Leinhardt, S. (1981). An exponential family of probability dis-

tributions for directed graphs. Journal of the American Statistical Association,

76(373):33–50.

Hoover, D. (1979). Relations on probability spaces and arrays of random variables.

Ji, P. and Jin, J. (2016). Coauthorship and citation networks for statisticians. Ann.

Appl. Stat., 10(4):1779–1812.

Kahle, M. (2017). Random simplicial complexes. In Handbook of Discrete and Com-

putational Geometry, chapter 22, pages 581–603. Chapman and Hall/CRC.

Kalman, R. E. (1960). A new approach to linear filtering and prediction problems.

Transactions of the ASME–Journal of Basic Engineering, 82(Series D):35–45.

Kantas, N., Doucet, A., Singh, S., and Maciejowski, J. (2009). An overview of sequen-

tial monte carlo methods for parameter estimation in general state-space models.

IFAC Proceedings Volumes, 42(10):774 – 785. 15th IFAC Symposium on System

Identification.

BIBLIOGRAPHY 152

Kantas, N., Doucet, A., Singh, S. S., Maciejowski, J., and Chopin, N. (2015). On

particle methods for parameter estimation in state-space models. Statist. Sci.,

30(3):328–351.

Karonski, M. and Luczak, T. (2002). The phase transition in a random hypergraph.

Journal of Computational and Applied Mathematics, 142(1):125 – 135. Probabilistic

Methods in Combinatorics and Combinatorial Optimization.

Karrer, B. and Newman, M. E. J. (2011). Stochastic blockmodels and community

structure in networks. Phys. Rev. E, 83:016107.

Ke, Z. T., Shi, F., and Xia, D. (2019). Community Detection for Hypergraph Networks

via Regularized Tensor Power Iteration. arXiv e-prints, page arXiv:1909.06503.

Khan, Z., Balch, T., and Dellaert, F. (2005). Mcmc-based particle filtering for tracking

a variable number of interacting targets. IEEE Transactions on Pattern Analysis

and Machine Intelligence, 27(11):1805–1819.

Kim, B., Lee, K. H., Xue, L., and Niu, X. (2018). A review of dynamic network

models with latent variables. Statist. Surv., 12:105–135.

Kim, C., Bandeira, A. S., and Goemans, M. X. (2018). Stochastic Block Model for

Hypergraphs: Statistical limits and a semidefinite programming approach. arXiv

e-prints, page arXiv:1807.02884.

Kitsak, M., Voitalov, I., and Krioukov, D. (2019). Link prediction with hyperbolic

geometry. arXiv e-prints, page arXiv:1903.08810.

Kolaczyk, E. D. (2009). Statistical Analysis of Network Data: Methods and Models.

Springer Publishing Company, Incorporated, 1st edition.

KONECT (2017). Corporate leadership network dataset. http://konect.

uni-koblenz.de/networks/brunson_corporate-leadership.

BIBLIOGRAPHY 153

Krioukov, D., Papadopoulos, F., Kitsak, M., Vahdat, A., and Boguna, M. (2010).

Hyperbolic geometry of complex networks. Phys. Rev. E, 82:036106.

Krivitsky, P. N., Handcock, M. S., Raftery, A. E., and Hoff, P. D. (2009). Representing

degree distributions, clustering, and homophily in social networks with latent cluster

random effects models. Social Networks, 31(3):204 – 213.

Kunegis, J. (2013). Konect: The koblenz network collection. In Proceedings of the

22Nd International Conference on World Wide Web, WWW ’13 Companion, pages

1343–1350, New York, NY, USA. ACM.

Lazarsfeld, P. and Henry, N. (1968). Latent structure analysis. Houghton, Mifflin.

Le, C. M., Levin, K., and Levina, E. (2018). Estimating a network from multiple

noisy realizations. Electron. J. Statist., 12(2):4697–4740.

Lee, W., McCormick, T. H., Neil, J., and Sodja, C. (2019). Anomaly detection in

large scale networks with latent space models.

Leskovec, J. and Krevl, A. (2014). SNAP Datasets: Stanford large network dataset

collection. http://snap.stanford.edu/data.

Lin, C., Chien, I. E., and Wang, I. (2017). On the fundamental statistical limit of com-

munity detection in random hypergraphs. In 2017 IEEE International Symposium

on Information Theory (ISIT), pages 2178–2182.

Liu, D., Blenn, N., and Mieghem, P. V. (2013). Characterising and modelling social

networks with overlapping communities. Int. J. Web Based Communities, 9(3):371–

391.

Liu, J. and West, M. (2001). Combined Parameter and State Estimation in

Simulation-Based Filtering, pages 197–223. Springer New York, New York, NY.

Lopes, H. F. and Tsay, R. S. (2011). Particle filters and bayesian inference in financial

econometrics. Journal of Forecasting, 30(1):168–209.

BIBLIOGRAPHY 154

Lovasz, L. (2012). Large Networks and Graph Limits., volume 60 of Colloquium

Publications. American Mathematical Society.

Ludkin, M., Eckley, I., and Neal, P. (2018). Dynamic stochastic block models: pa-

rameter estimation and detection of changes in community structure. Statistics and

Computing, 28(6):1201–1213.

Lunagomez, S., Mukherjee, S., Wolpert, R. L., and Airoldi, E. M. (2017). Geomet-

ric representations of random hypergraphs. Journal of the American Statistical

Association, 112(517):363–383.

Lunagomez, S., Olhede, S. C., and Wolfe, P. J. (2019). Modeling Network Populations

via Graph Distances. arXiv e-prints, page arXiv:1904.07367.

Mansilla, S. P. and Serra, O. (2008). On s-arc transitive hypergraphs. European

Journal of Combinatorics, 29(4):1003 – 1011. Homomorphisms: Structure and

Highlights.

Marin, J.-M., Pudlo, P., Robert, C. P., and Ryder, R. J. (2012). Approximate bayesian

computational methods. Statistics and Computing, 22(6):1167–1180.

McCormick, T. H. and Zheng, T. (2015). Latent surface models for networks us-

ing aggregated relational data. Journal of the American Statistical Association,

110(512):1684–1695.

Naesseth, C. A., Lindsten, F., and Schon, T. B. (2015). Nested sequential monte

carlo methods. In Proceedings of the 32nd International Conference on Interna-

tional Conference on Machine Learning - Volume 37, ICML’15, pages 1292–1301.

JMLR.org.

Neal, R. M. (2001). Annealed importance sampling. Statistics and Computing,

11(2):125–139.

BIBLIOGRAPHY 155

Nemeth, C., Fearnhead, P., and Mihaylova, L. (2016). Particle approximations of

the score and observed information matrix for parameter estimation in statespace

models with linear computational cost. Journal of Computational and Graphical

Statistics, 25(4):1138–1157.

Newman, M. E. J. (2010). Networks: an introduction. Oxford University Press,

Oxford; New York.

Ng, T. L. J. and Murphy, T. B. (2018). Model-based clustering for random hyper-

graphs. ArXiv e-prints.

Ng, T. L. J. and Murphy, T. B. (2019). Generalized random dot product graph.

Nickel, C. (2007). Random Dot Product Graphs: A Model for Social Networks. PhD

thesis, Department of Applied Mathematics and Statistics, The Johns Hopkins

University.

Nowicki, K. and Snijders, T. A. B. (2001). Estimation and prediction for stochastic

blockstructures. Journal of the American Statistical Association, 96(455):1077–

1087.

Park, J. and Ionides, E. L. (2019). Inference on high-dimensional implicit dy-

namic models using a guided intermediate resampling filter. arXiv e-prints, page

arXiv:1708.08543.

Park, J. H. and Sohn, Y. (2018). Detecting structural changes in longitudinal network

data. Bayesian Anal. Advance publication.

Patcha, A. and Park, J.-M. (2007). An overview of anomaly detection techniques:

Existing solutions and latest technological trends. Computer Networks, 51(12):3448

– 3470.

Penrose, M. (2003). Random Geometric Graphs. Oxford Studies in Probability.

BIBLIOGRAPHY 156

Petersen, K. B. and Pedersen, M. S. (2012). The matrix cookbook. Version 20121115.

Pitt, M. K. and Shephard, N. (1999). Filtering via simulation: Auxiliary particle

filters. Journal of the American Statistical Association, 94(446):590–599.

Polson, N. G., Scott, J. G., and Windle, J. (2013). Bayesian inference for logistic

models using polya-gamma latent variables. Journal of the American Statistical

Association, 108(504):1339–1349.

Polson, N. G., Stroud, J. R., and Mller, P. (2008). Practical filtering with sequential

parameter learning. Journal of the Royal Statistical Society: Series B (Statistical

Methodology), 70(2):413–428.

Poyiadjis, G., Doucet, A., and Singh, S. S. (2011). Particle approximations of the

score and observed information matrix in state space models with application to

parameter estimation. Biometrika, 98(1):65–80.

Pronzato, L., Wynn, H. P., and Zhigljavsky, A. (2019). Bregman divergences based

on optimal design criteria and simplicial measures of dispersion. Statistical Papers,

60(2):195–214.

Pronzato, L., Wynn, H. P., and Zhigljavsky, A. A. (2018). Simplicial variances,

potentials and mahalanobis distances. Journal of Multivariate Analysis, 168:276 –

289.

Raftery, A. E., Niu, X., Hoff, P. D., and Yeung, K. Y. (2012). Fast inference for the

latent space network model using a case-control approximate likelihood. Journal of

Computational and Graphical Statistics, 21(4):901–919.

Rastelli, R., Friel, N., and Raftery, A. E. (2016). Properties of latent variable network

models. Network Science, 4(4):407432.

Rastelli, R., Maire, F., and Friel, N. (2018). Computationally efficient inference for

latent position network models. arXiv e-prints, page arXiv:1804.02274.

BIBLIOGRAPHY 157

Rebeschini, P. and van Handel, R. (2015). Can local particle filters beat the curse of

dimensionality? Ann. Appl. Probab., 25(5):2809–2866.

Rubin-Delanchy, P., Priebe, C. E., and Tang, M. (2017). Consistency of adjacency

spectral embedding for the mixed membership stochastic blockmodel.

Rubin-Delanchy, P., Priebe, C. E., Tang, M., and Cape, J. (2017). A statistical

interpretation of spectral embedding: the generalised random dot product graph.

arXiv e-prints, page arXiv:1709.05506.

Salnikov, V., Cassese, D., and Lambiotte, R. (2018). Simplicial complexes and com-

plex systems. European Journal of Physics, 40(1):014001.

Salter-Townshend, M. and McCormick, T. H. (2017). Latent space models for multi-

view network data. Ann. Appl. Stat., 11(3):1217–1244.

Salter-Townshend, M. and Murphy, T. B. (2013). Variational bayesian inference for

the latent position cluster model for network data. Computational Statistics and

Data Analysis, 57(1):661 – 671.

Salter-Townshend, M., White, A., Gollini, I., and Murphy, T. B. (2012). Review of

statistical network analysis: models, algorithms, and software. Statistical Analysis

and Data Mining: The ASA Data Science Journal, 5(4):243–264.

Sarkar, P. and Moore, A. W. (2006). Dynamic social network analysis using latent

space models. In Weiss, Y., Scholkopf, B., and Platt, J. C., editors, Advances in

Neural Information Processing Systems 18, pages 1145–1152. MIT Press.

Septier, F., Pang, S. K., Carmi, A., and Godsill, S. (2009). On mcmc-based particle

methods for bayesian filtering: Application to multitarget tracking. In 2009 3rd

IEEE International Workshop on Computational Advances in Multi-Sensor Adap-

tive Processing (CAMSAP), pages 360–363.

BIBLIOGRAPHY 158

Septier, F. and Peters, G. W. (2015). An Overview of Recent Advances in Monte-Carlo

Methods for Bayesian Filtering in High-Dimensional Spaces. In Peters, G. W. and

Matsui, T., editors, Theoretical Aspects of Spatial-Temporal Modeling. Springer-

Briefs - JSS Research Series in Statistics.

Sewell, D. K. and Chen, Y. (2015a). Analysis of the formation of the structure of

social networks by using latent space models for ranked dynamic networks. Journal

of the Royal Statistical Society: Series C (Applied Statistics), 64(4):611–633.

Sewell, D. K. and Chen, Y. (2015b). Latent space models for dynamic networks.

Journal of the American Statistical Association, 110(512):1646–1657.

Sewell, D. K. and Chen, Y. (2016). Latent space models for dynamic networks with

weighted edges. Social Networks, 44:105 – 116.

Sewell, D. K. and Chen, Y. (2017). Latent space approaches to community detection

in dynamic networks. Bayesian Anal., 12(2):351–377.

Sharma, A., Srivastava, J., and Chandra, A. (2014). Predicting Multi-actor collabo-

rations using Hypergraphs. arXiv e-prints, page arXiv:1401.6404.

Smith, A. L., Asta, D. M., and Calder, C. A. (2017). The Geometry of Continuous

Latent Space Models for Network Data. ArXiv e-prints.

Snyder, C., Bengtsson, T., Bickel, P., and Anderson, J. (2008). Obstacles to high-

dimensional particle filtering. Monthly Weather Review, 136(12):4629–4640.

Snyder, C., Bengtsson, T., and Morzfeld, M. (2015). Performance bounds for particle

filters using the optimal proposal. Monthly Weather Review, 143(11):4750–4761.

Spencer, N. A. and Rohilla Shalizi, C. (2017). Projective, Sparse, and Learnable

Latent Position Network Models. arXiv e-prints, page arXiv:1709.09702.

Stasi, D., Sadeghi, K., Rinaldo, A., Petrovic, S., and Fienberg, S. E. (2014). β models

for random hypergraphs with a given degree sequence. ArXiv e-prints.

BIBLIOGRAPHY 159

Stehle, J., Voirin, N., Barrat, A., Cattuto, C., Isella, L., Pinton, J.-F., Quaggiotto, M.,

Van den Broeck, W., Regis, C., Lina, B., and Vanhems, P. (2011). High-resolution

measurements of face-to-face contact patterns in a primary school. PLOS ONE,

6(8):1–13.

Storvik, G. (2002). Particle filters for state-space models with the presence of unknown

static parameters. IEEE Transactions on Signal Processing, 50(2):281–289.

Sweet, T. M., Thomas, A. C., and Junker, B. W. (2013). Hierarchical network models

for education research: Hierarchical latent space models. Journal of Educational

and Behavioral Statistics, 38(3):295–318.

Teerapabolarn, K. (2014). An improvement of poisson approximation for sums of

dependent bernoulli random variables. Communications in Statistics - Theory and

Methods, 43(8):1758–1777.

The GUDHI Project (2015). GUDHI User and Reference Manual. GUDHI Editorial

Board.

Truong, C., Oudre, L., and Vayatis, N. (2018). Selective review of offline change point

detection methods.

van Duijn, M. A. J., Snijders, T. A. B., and Zijlstra, B. J. H. (2004). p2: a random

effects model with covariates for directed graphs. Statistica Neerlandica, 58(2):234–

254.

Vietoris, L. (1927). Uber den hheren zusammenhang kompakter raume und eine klasse

von zusammenhangstreuen abbildungen. Mathematische Annalen, pages 454–472.

Wang, Y., Chakrabarti, A., Sivakoff, D., and Parthasarathy, S. (2017). Hierarchical

change point detection on dynamic networks. In Proceedings of the 2017 ACM on

Web Science Conference, WebSci ’17, pages 171–179, New York, NY, USA. ACM.

BIBLIOGRAPHY 160

Ward, M. D., Ahlquist, J. S., and Rozenas, A. (2013). Gravity’s rainbow: A dynamic

latent space model for the world trade network. Network Science, 1(1):95118.

Wasserman, S. and Pattison, P. (1996). Logit models and logistic regressions for social

networks: I. an introduction to markov graphs andp. Psychometrika, 61(3):401–425.

Young, S. J. and Scheinerman, E. R. (2007). Random dot product graph models for

social networks. In Bonato, A. and Chung, F. R. K., editors, Algorithms and Models

for the Web-Graph, pages 138–149, Berlin, Heidelberg. Springer Berlin Heidelberg.

Zachary, W. (1977). An information flow model for conflict and fission in small groups.

Journal of Anthropological Research, 33:452–473.

Zhou, D., Huang, J., and Scholkopf, B. (2006). Learning with hypergraphs: Clus-

tering, classification, and embedding. In Proceedings of the 19th International

Conference on Neural Information Processing Systems, NIPS’06, pages 1601–1608,

Cambridge, MA, USA. MIT Press.

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